Theory of Computing Report

Authors: Ethan Dickey, Athina Terzoglou

The tournament equilibrium set (TEQ), proposed by Schwartz in 1990, is a classical voting rule based on pairwise majority comparisons. Its definition is recursive: an alternative is judged by the alternatives that beat it, called its dominators, and by which of those dominators TEQ itself selects. Deciding whether a given alternative belongs to TEQ was known to be NP-hard and to lie in PSPACE. We close this gap by proving that the problem is PSPACE-complete. To our knowledge, this is the first PSPACE-completeness result for membership in a tournament solution. The hardness persists even when every nonempty subtournament has a unique minimal retentive set, the property conjectured by Schwartz to hold for all tournaments. Moreover, every such subtournament in our hard instances contains an alternative with at most three dominators. We prove that a single alternative with at most four dominators suffices both to guarantee uniqueness and to find a TEQ winner in $O(N^2)$ time. Thus, on our hard instances, finding one winner is efficient even though recognizing a specified winner is PSPACE-complete. We also study how the complexity changes with the dominator bound. If every nonempty subtournament has an alternative with at most one dominator, the entire TEQ can be computed in quadratic time; allowing two dominators makes membership NP-hard. Under the same condition with four dominators, all $t$ winners can be computed in $O(2^t N^2)$ time, where $N$ is the number of alternatives.

Authors: Ethan Dickey, Athina Terzoglou

The tournament equilibrium set (TEQ), proposed by Schwartz in 1990, is a classical voting rule based on pairwise majority comparisons. Its definition is recursive: an alternative is judged by the alternatives that beat it, called its dominators, and by which of those dominators TEQ itself selects. Deciding whether a given alternative belongs to TEQ was known to be NP-hard and to lie in PSPACE. We close this gap by proving that the problem is PSPACE-complete. To our knowledge, this is the first PSPACE-completeness result for membership in a tournament solution. The hardness persists even when every nonempty subtournament has a unique minimal retentive set, the property conjectured by Schwartz to hold for all tournaments. Moreover, every such subtournament in our hard instances contains an alternative with at most three dominators. We prove that a single alternative with at most four dominators suffices both to guarantee uniqueness and to find a TEQ winner in $O(N^2)$ time. Thus, on our hard instances, finding one winner is efficient even though recognizing a specified winner is PSPACE-complete. We also study how the complexity changes with the dominator bound. If every nonempty subtournament has an alternative with at most one dominator, the entire TEQ can be computed in quadratic time; allowing two dominators makes membership NP-hard. Under the same condition with four dominators, all $t$ winners can be computed in $O(2^t N^2)$ time, where $N$ is the number of alternatives.

Friday, November 13 2026, 01:00

Sowing the Seeds of Intractability: Kalah is PSPACE-complete

Authors: Ethan Dickey, Ethan Mader

Kalah is a classic two-player game in the mancala family of sowing games. Computers have solved several standard-size starting positions, but the complexity of the game on boards of growing size has remained open. We study the natural generalization in which the number of pits per side is part of the input and an arbitrary position is given, with the stone counts given in unary. We show that deciding whether a designated player can force a win from such a position is PSPACE-complete. The hard positions have Alice to move first and both stores initially empty. The same positions are hard under either common convention for capture opposite an empty pit, and with or without the bonus rule, which gives a player another turn when the last stone is sown into their own store. We introduce Line Kalah as an intermediate problem, a game on a line in which stones that leave the line effectively disappear and Alice must save each of her target pits before Bob captures it, and we reduce quantified Boolean formulas to it. The reduction uses a set of sequential timer piles whose last stones approach a group of targets, forcing Alice to choose between two captures in order to set the truth value of a variable, and a final capture gadget to test a clause chosen by Bob (only one is needed adversarially). Since either player may move any pile at any time, the stages are placed so that premature timer firings protect earlier potential targets. For a true formula, Alice's strategy against arbitrary opposing play either gains enough time to save the remaining targets (after Bob deviates from the intended play) or reaches the final test with a correctly encoded, protected input. Finally, the line positions are embedded in Kalah with both stores empty. A separate Bob pit holds the score adjustment, so the winner takes a strict majority of the stones.

Authors: Ethan Dickey, Ethan Mader

Kalah is a classic two-player game in the mancala family of sowing games. Computers have solved several standard-size starting positions, but the complexity of the game on boards of growing size has remained open. We study the natural generalization in which the number of pits per side is part of the input and an arbitrary position is given, with the stone counts given in unary. We show that deciding whether a designated player can force a win from such a position is PSPACE-complete. The hard positions have Alice to move first and both stores initially empty. The same positions are hard under either common convention for capture opposite an empty pit, and with or without the bonus rule, which gives a player another turn when the last stone is sown into their own store. We introduce Line Kalah as an intermediate problem, a game on a line in which stones that leave the line effectively disappear and Alice must save each of her target pits before Bob captures it, and we reduce quantified Boolean formulas to it. The reduction uses a set of sequential timer piles whose last stones approach a group of targets, forcing Alice to choose between two captures in order to set the truth value of a variable, and a final capture gadget to test a clause chosen by Bob (only one is needed adversarially). Since either player may move any pile at any time, the stages are placed so that premature timer firings protect earlier potential targets. For a true formula, Alice's strategy against arbitrary opposing play either gains enough time to save the remaining targets (after Bob deviates from the intended play) or reaches the final test with a correctly encoded, protected input. Finally, the line positions are embedded in Kalah with both stores empty. A separate Bob pit holds the score adjustment, so the winner takes a strict majority of the stones.

Friday, November 13 2026, 01:00

Computation with Backward Feedback: Power, Observation, and Cost

Authors: Saman Farhat, Farzad Didehvar

How much can a computation gain by passing information back to an earlier stage, and how soon can an observer obtain the answer? We study deterministic programs whose runs are accessed at randomly selected stages. A program keeps a register, a finite block of memory. A random clock selects the current stage; each tick takes a random step, with a law fixed independently of input, program and register. When the clock moves backward, the register contents at the stage it leaves start a new run, and the clock continues where it lands. Equilibrium readout requires a distribution of clock position and register that one tick leaves unchanged, and a correct final answer under every such distribution. For bounded steps in both directions with negative average, we obtain an exact classification at each polynomial scale s. If running time and register length are both O(s), feedback gives exactly nondeterministic space O(s); without transmission the class is deterministic time O(s), which is strictly smaller. An observer that starts at stage zero with an all-zero register, reads sequentially and never accepts a nonmember sees, with fewer ticks than any exponential, only languages fixed by a bounded input prefix. Exponentially many ticks reveal every linear-time language. For a clock moving one step up or down, we bound the wait for useful backward moves and the rate of random erasures a stored counter tolerates. The results separate the power of feedback at equilibrium, observation from a fixed start, and the cost of obtaining and keeping information.

Authors: Saman Farhat, Farzad Didehvar

How much can a computation gain by passing information back to an earlier stage, and how soon can an observer obtain the answer? We study deterministic programs whose runs are accessed at randomly selected stages. A program keeps a register, a finite block of memory. A random clock selects the current stage; each tick takes a random step, with a law fixed independently of input, program and register. When the clock moves backward, the register contents at the stage it leaves start a new run, and the clock continues where it lands. Equilibrium readout requires a distribution of clock position and register that one tick leaves unchanged, and a correct final answer under every such distribution. For bounded steps in both directions with negative average, we obtain an exact classification at each polynomial scale s. If running time and register length are both O(s), feedback gives exactly nondeterministic space O(s); without transmission the class is deterministic time O(s), which is strictly smaller. An observer that starts at stage zero with an all-zero register, reads sequentially and never accepts a nonmember sees, with fewer ticks than any exponential, only languages fixed by a bounded input prefix. Exponentially many ticks reveal every linear-time language. For a clock moving one step up or down, we bound the wait for useful backward moves and the rate of random erasures a stored counter tolerates. The results separate the power of feedback at equilibrium, observation from a fixed start, and the cost of obtaining and keeping information.

Friday, November 13 2026, 01:00

Topological Entropy of Group Cellular Automata is Effectively Computable

Authors: Niccolò Castronuovo, Alberto Dennunzio, Jarkko Kari, Luciano Margara

Topological entropy is a fundamental invariant of dynamical systems, quantifying their complexity. For cellular automata, however, determining topological entropy is generally algorithmically intractable. We study group cellular automata, namely cellular automata over a finite group alphabet whose global map is a group endomorphism. In dimension one, we decompose topological entropy into two one-sided components describing information propagating from the left and from the right in space--time. We introduce finite families of left and right traces and prove that the entropy is the sum of their growth rates. Hence, the entropy is completely determined by the growth rate of a fixed-width family of space--time patterns. This reduction from arbitrarily large windows to uniformly bounded width yields an effective procedure for computing the entropy. In particular, the entropy of every one-dimensional group cellular automaton is the logarithm of a positive integer. Using Pontryagin duality and the Bridge Theorem between topological and algebraic entropy, we also derive an explicit formula for linear cellular automata over $(\mathbb{Z}/p\mathbb{Z})^n$ in terms of their characteristic polynomial. Moreover, in every dimension $D\geq 2$, a group cellular automaton is either equicontinuous with zero entropy or sensitive to initial conditions with infinite entropy. Combined with decidability of equicontinuity, this implies that topological entropy is effectively computable from the local rule in every dimension. Thus, group cellular automata form a natural nontrivial class in which algebraic structure restores computability of topological entropy.

Authors: Niccolò Castronuovo, Alberto Dennunzio, Jarkko Kari, Luciano Margara

Topological entropy is a fundamental invariant of dynamical systems, quantifying their complexity. For cellular automata, however, determining topological entropy is generally algorithmically intractable. We study group cellular automata, namely cellular automata over a finite group alphabet whose global map is a group endomorphism. In dimension one, we decompose topological entropy into two one-sided components describing information propagating from the left and from the right in space--time. We introduce finite families of left and right traces and prove that the entropy is the sum of their growth rates. Hence, the entropy is completely determined by the growth rate of a fixed-width family of space--time patterns. This reduction from arbitrarily large windows to uniformly bounded width yields an effective procedure for computing the entropy. In particular, the entropy of every one-dimensional group cellular automaton is the logarithm of a positive integer. Using Pontryagin duality and the Bridge Theorem between topological and algebraic entropy, we also derive an explicit formula for linear cellular automata over $(\mathbb{Z}/p\mathbb{Z})^n$ in terms of their characteristic polynomial. Moreover, in every dimension $D\geq 2$, a group cellular automaton is either equicontinuous with zero entropy or sensitive to initial conditions with infinite entropy. Combined with decidability of equicontinuity, this implies that topological entropy is effectively computable from the local rule in every dimension. Thus, group cellular automata form a natural nontrivial class in which algebraic structure restores computability of topological entropy.

Friday, November 13 2026, 01:00

Exact Quantum Algorithms for Composed Boolean Functions

Authors: Gagan Chiruvella, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh

We study exact quantum algorithms for block compositions of total Boolean functions. For an outer function $f$ of degree $D$ and an inner function $g$ on $m$ bits, we prove $$ Q_E(f\circ g)=O\!\left(mD+Q_E(f)Q_{1/(4D)}(g)\right). $$ The algorithm combines approximate inner evaluations with a finite exact correction. Its two costs are governed by different parameters of the outer function: exact query complexity determines the cost of simulation, whereas polynomial degree determines the number of inner blocks used in the correction. We first establish the construction with an arity-dependent error threshold and then obtain the degree-dependent threshold above. For recursive ternary not-all-equal, this gives an exact algorithm using $(3/\sqrt2)^{k(1+o(1))}$ queries asymptotically. More generally, for every fixed nonconstant total gadget $h$, we determine the exact recursive rate as $\max\{\textrm{deg}(h),\operatorname{Adv}^{\pm}(h)\}$ and prove $Q_E(h^{\circ k})=\widetilde O_h(R(h^{\circ k})^2)$. The composition limit also allows to transfer separations from $Q_{\varepsilon}$ and $°$ to exact quantum query complexity $Q_E$ (similar to fractional block sensitivity to block sensitivity transfer in Tal and Kulkarni 2016, Chicago Journal of Theoretical Computer Science). We construct a family with $s=\operatorname{bs}=\operatorname{fbs}\ge Q_E^{3/2-o(1)}$, improving the previous exponent $1.1527\ldots$. A second construction gives $R \ge Q_E^{2-o(1)}$, improving the previous exponent $3/2$. We also allow different inner functions and give analogous quantitative separations in quantum communication.

Authors: Gagan Chiruvella, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh

We study exact quantum algorithms for block compositions of total Boolean functions. For an outer function $f$ of degree $D$ and an inner function $g$ on $m$ bits, we prove $$ Q_E(f\circ g)=O\!\left(mD+Q_E(f)Q_{1/(4D)}(g)\right). $$ The algorithm combines approximate inner evaluations with a finite exact correction. Its two costs are governed by different parameters of the outer function: exact query complexity determines the cost of simulation, whereas polynomial degree determines the number of inner blocks used in the correction. We first establish the construction with an arity-dependent error threshold and then obtain the degree-dependent threshold above. For recursive ternary not-all-equal, this gives an exact algorithm using $(3/\sqrt2)^{k(1+o(1))}$ queries asymptotically. More generally, for every fixed nonconstant total gadget $h$, we determine the exact recursive rate as $\max\{\textrm{deg}(h),\operatorname{Adv}^{\pm}(h)\}$ and prove $Q_E(h^{\circ k})=\widetilde O_h(R(h^{\circ k})^2)$. The composition limit also allows to transfer separations from $Q_{\varepsilon}$ and $°$ to exact quantum query complexity $Q_E$ (similar to fractional block sensitivity to block sensitivity transfer in Tal and Kulkarni 2016, Chicago Journal of Theoretical Computer Science). We construct a family with $s=\operatorname{bs}=\operatorname{fbs}\ge Q_E^{3/2-o(1)}$, improving the previous exponent $1.1527\ldots$. A second construction gives $R \ge Q_E^{2-o(1)}$, improving the previous exponent $3/2$. We also allow different inner functions and give analogous quantitative separations in quantum communication.

Friday, November 13 2026, 01:00

Weak Composition Theorems for Randomized Communication

Authors: Pachara Sawettamalya

We present two results towards the randomized communication complexity of composed functions. Our contributions include: - \textbf{Lifting-by-Influence.} For any binary function $f$ and any outer gadget $g:\{0,1\}^n \rightarrow \mathbb{R}$, we show that up to additive terms, computing $g \circ f^n$ information-theoretically requires solving at least $\operatorname{Inf}(g)$ instances of $f$, where $\operatorname{Inf}(g)$ denotes the total influence of $g$. To our knowledge, this is the first composition lower bound for randomized communication that applies to \emph{every} choice of $f$ and $g$. - \textbf{Symmetric Composition.} When $g$ is a symmetric function with a value transition at Hamming weight $d$, we strengthen the number of required instances to $\min\{d,n+1-d\}$. As an implication, this recovers the previously known direct-sum theorem, XOR lemma, and majority lemma. Our proofs develop novel and distinctive usages of the classical \emph{input embedding} argument of Barak, Braverman, Chen, and Rao (SICOMP 2013). As an additional ingredient, we introduce a new public sampling framework that avoids auxiliary communication during the recovery phase of the protocol, ultimately reducing the extra cost from $O(n)$ bits of information to just $O(1)$.

Authors: Pachara Sawettamalya

We present two results towards the randomized communication complexity of composed functions. Our contributions include: - \textbf{Lifting-by-Influence.} For any binary function $f$ and any outer gadget $g:\{0,1\}^n \rightarrow \mathbb{R}$, we show that up to additive terms, computing $g \circ f^n$ information-theoretically requires solving at least $\operatorname{Inf}(g)$ instances of $f$, where $\operatorname{Inf}(g)$ denotes the total influence of $g$. To our knowledge, this is the first composition lower bound for randomized communication that applies to \emph{every} choice of $f$ and $g$. - \textbf{Symmetric Composition.} When $g$ is a symmetric function with a value transition at Hamming weight $d$, we strengthen the number of required instances to $\min\{d,n+1-d\}$. As an implication, this recovers the previously known direct-sum theorem, XOR lemma, and majority lemma. Our proofs develop novel and distinctive usages of the classical \emph{input embedding} argument of Barak, Braverman, Chen, and Rao (SICOMP 2013). As an additional ingredient, we introduce a new public sampling framework that avoids auxiliary communication during the recovery phase of the protocol, ultimately reducing the extra cost from $O(n)$ bits of information to just $O(1)$.

Friday, November 13 2026, 01:00

The Ultimate Karp-Lipton Collapse

Authors: Edward A. Hirsch, Ilya Volkovich

Karp and Lipton proved that if $NP\subseteq P/poly$, then $PH$ collapses. A long line of work pushed the collapse to smaller classes. In particular, Chakaravarthy and Roy obtained two such collapses, to (apparently) incomparable classes $O_2^P$ and $P^{prMA}$. We united them earlier by a collapse to the input-oblivious version $P^{prOMA}$ of $P^{prMA}$, which is contained in both; and, jointly with Williams, to a slightly smaller class. Two other deepest known collapses are to $ZPP^{ONP}$ (following from Köbler and Watanabe) and to $OL_2^P$ (proved in this paper). We unite the three collapses by a collapse to a class contained in all of them. We use a three-stage computational model OGTD (oblivious generator-tester-decoder): a polynomial-time generator with an oracle computes an advice string from $1^n$, a tester certifies it, and a polynomial-time decoder uses it (it is the only one that sees the actual input). Let $OM$ be the class of OGTD languages defined by a restricted version of $prOMA$ as the generator's oracle, by a co-$ONP$ tester, and a $P$ decoder. We show that it is contained in the three collapse classes above and prove that $PH$ collapses to $OM$ under $NP\subseteq P/poly$. The collapse also holds for an unambiguous version of $OM$. All these input-oblivious collapse classes lie in $P^{prSBP}$, and so do even the non-input-oblivious versions $P^{prMA}$, $L_2^P$ and $S_2^P$: we prove that $P^{prS_2^P}\subseteq P^{prSBP}$. This strengthens the containment $S_2^P\subseteq P^{prAM}$ of Chakaravarthy and Roy and our earlier containment $L_2^P\subseteq P^{prSBP}$. The proof unites the Goldreich-Wigderson-type procedure of Chakaravarthy and Roy with our earlier approximate-counting technique. Its input-oblivious counterpart concerns $O_2^P$, the first collapse class of Chakaravarthy and Roy: every language in $O_2^P$ has polynomial-size circuits constructible in $FP^{prSBP}$.

Authors: Edward A. Hirsch, Ilya Volkovich

Karp and Lipton proved that if $NP\subseteq P/poly$, then $PH$ collapses. A long line of work pushed the collapse to smaller classes. In particular, Chakaravarthy and Roy obtained two such collapses, to (apparently) incomparable classes $O_2^P$ and $P^{prMA}$. We united them earlier by a collapse to the input-oblivious version $P^{prOMA}$ of $P^{prMA}$, which is contained in both; and, jointly with Williams, to a slightly smaller class. Two other deepest known collapses are to $ZPP^{ONP}$ (following from Köbler and Watanabe) and to $OL_2^P$ (proved in this paper). We unite the three collapses by a collapse to a class contained in all of them. We use a three-stage computational model OGTD (oblivious generator-tester-decoder): a polynomial-time generator with an oracle computes an advice string from $1^n$, a tester certifies it, and a polynomial-time decoder uses it (it is the only one that sees the actual input). Let $OM$ be the class of OGTD languages defined by a restricted version of $prOMA$ as the generator's oracle, by a co-$ONP$ tester, and a $P$ decoder. We show that it is contained in the three collapse classes above and prove that $PH$ collapses to $OM$ under $NP\subseteq P/poly$. The collapse also holds for an unambiguous version of $OM$. All these input-oblivious collapse classes lie in $P^{prSBP}$, and so do even the non-input-oblivious versions $P^{prMA}$, $L_2^P$ and $S_2^P$: we prove that $P^{prS_2^P}\subseteq P^{prSBP}$. This strengthens the containment $S_2^P\subseteq P^{prAM}$ of Chakaravarthy and Roy and our earlier containment $L_2^P\subseteq P^{prSBP}$. The proof unites the Goldreich-Wigderson-type procedure of Chakaravarthy and Roy with our earlier approximate-counting technique. Its input-oblivious counterpart concerns $O_2^P$, the first collapse class of Chakaravarthy and Roy: every language in $O_2^P$ has polynomial-size circuits constructible in $FP^{prSBP}$.

Friday, November 13 2026, 01:00

The multiplicative complexity of the three-bit low-half multiplier

Authors: Bshara Alhodali, Gregory Morse

We prove that the multiplicative complexity of the three-bit low-half multiplier is six, giving a structural, enumeration-free lower-bound proof valid against unrestricted nonlinear reuse in Boolean XOR-AND circuits. The value was previously established by SAT searches of the second author. An explicit six-AND circuit supplies the upper bound. The lower bound combines derivatives, quadratic products, complementary minors, and extraction modulo a Boolean ideal. Independent finite checks accompany the arguments. $\mathrm{MC}(M_3)=6$ is fully formalized in Lean 4. The kernel-checked proof uses only the axioms propext, Classical.choice and Quot.sound.

Authors: Bshara Alhodali, Gregory Morse

We prove that the multiplicative complexity of the three-bit low-half multiplier is six, giving a structural, enumeration-free lower-bound proof valid against unrestricted nonlinear reuse in Boolean XOR-AND circuits. The value was previously established by SAT searches of the second author. An explicit six-AND circuit supplies the upper bound. The lower bound combines derivatives, quadratic products, complementary minors, and extraction modulo a Boolean ideal. Independent finite checks accompany the arguments. $\mathrm{MC}(M_3)=6$ is fully formalized in Lean 4. The kernel-checked proof uses only the axioms propext, Classical.choice and Quot.sound.

Friday, November 13 2026, 01:00

Sheaf-Inspired Structured Scoring Extensions of LightGCN for Collaborative Filtering

Authors: Ksenia Belkovskaya, Timur Lidzhiev, Maria Stebliy, Alexander Tarakanov

Graph-based collaborative filtering learns user and item representations by propagating embeddings over a user--item interaction graph. We investigate a modular, sheaf-inspired scoring extension placed on top of a three-layer LightGCN backbone. The motivation comes from cellular sheaves, where local vector spaces are related by linear restriction maps; here, this idea motivates structured, pair-dependent transformations of user and item embeddings rather than a change to graph propagation. We compare the identity inner-product scorer, diagonal, rank-one low-rank, and full matrix residual scorers, and a parameter-matched residual MLP baseline (\texttt{mlp\_full}) on Amazon-Book, Gowalla, MovieLens-1M, and Yelp2018. We report ranking quality and catalog coverage at cutoffs 10 and 20. The selected structured variants show modest, dataset-dependent changes in ranking quality and sometimes substantial changes in coverage. Our method is inspired by sheaf geometry but does not construct a cellular sheaf or implement sheaf-Laplacian diffusion; the experiments therefore do not establish a causal benefit from sheaf geometry itself.

Authors: Ksenia Belkovskaya, Timur Lidzhiev, Maria Stebliy, Alexander Tarakanov

Graph-based collaborative filtering learns user and item representations by propagating embeddings over a user--item interaction graph. We investigate a modular, sheaf-inspired scoring extension placed on top of a three-layer LightGCN backbone. The motivation comes from cellular sheaves, where local vector spaces are related by linear restriction maps; here, this idea motivates structured, pair-dependent transformations of user and item embeddings rather than a change to graph propagation. We compare the identity inner-product scorer, diagonal, rank-one low-rank, and full matrix residual scorers, and a parameter-matched residual MLP baseline (\texttt{mlp\_full}) on Amazon-Book, Gowalla, MovieLens-1M, and Yelp2018. We report ranking quality and catalog coverage at cutoffs 10 and 20. The selected structured variants show modest, dataset-dependent changes in ranking quality and sometimes substantial changes in coverage. Our method is inspired by sheaf geometry but does not construct a cellular sheaf or implement sheaf-Laplacian diffusion; the experiments therefore do not establish a causal benefit from sheaf geometry itself.

Friday, November 13 2026, 01:00

Asymptotic Dimension of Persistence Modules of Bounded Presentation

Authors: Sushovan Majhi, Atish Mitra, Žiga Virk

We compute the asymptotic dimension of spaces of multiparameter persistence modules of bounded presentation under the interleaving distance. The $d$-parameter modules with at most $n$ generators and $m$ relations have asymptotic dimension, and Assouad-Nagata dimension, exactly $d(n+m)$, the number of coordinates of their generator and relation grades, over every field; for two parameters this is $2(n+m)$. The space has continuous moduli, already for two parameters and $n,m\ge2$: over an infinite field it contains, at every scale, infinite uniformly separated families of pairwise non-isomorphic modules with one fibered barcode, so it is not doubling even at a fixed fibered barcode; already for $m\ge1$ it is not coarsely equivalent to a proper metric space. The proof is an induction on the number of distinct grade coordinates: an interleaving small compared with the gaps between these coordinates moves each of them by at most twice its size and preserves the induced grid representation, so the continuous moduli are invisible at large scale. Each class of modules at finite distance therefore embeds coarsely into Hilbert space; for $n,m\ge2$ over an uncountable field, the classes that contain these families admit no such embedding into a separable Hilbert space. Under the two-parameter matching distance $2(n+m)$ is a lower bound, attained for one generator and for two generators with at most one relation; on a finite net of $S$ slices the asymptotic dimension is at most $2nS$.

Authors: Sushovan Majhi, Atish Mitra, Žiga Virk

We compute the asymptotic dimension of spaces of multiparameter persistence modules of bounded presentation under the interleaving distance. The $d$-parameter modules with at most $n$ generators and $m$ relations have asymptotic dimension, and Assouad-Nagata dimension, exactly $d(n+m)$, the number of coordinates of their generator and relation grades, over every field; for two parameters this is $2(n+m)$. The space has continuous moduli, already for two parameters and $n,m\ge2$: over an infinite field it contains, at every scale, infinite uniformly separated families of pairwise non-isomorphic modules with one fibered barcode, so it is not doubling even at a fixed fibered barcode; already for $m\ge1$ it is not coarsely equivalent to a proper metric space. The proof is an induction on the number of distinct grade coordinates: an interleaving small compared with the gaps between these coordinates moves each of them by at most twice its size and preserves the induced grid representation, so the continuous moduli are invisible at large scale. Each class of modules at finite distance therefore embeds coarsely into Hilbert space; for $n,m\ge2$ over an uncountable field, the classes that contain these families admit no such embedding into a separable Hilbert space. Under the two-parameter matching distance $2(n+m)$ is a lower bound, attained for one generator and for two generators with at most one relation; on a finite net of $S$ slices the asymptotic dimension is at most $2nS$.

Friday, November 13 2026, 01:00

Optimal Polyomino Fences

Authors: Alexis Langlois-Rémillard, Mia N. Müßig, Helmut Podhaisky, Erika Roldan, Günter Rote, Maximilian Brömme, Matthes Norden, Maximilian Prietzel, Daniel Yu, Mykhailo Lyader

The $n$-omino fence problem asks how many cells can be surrounded by placing every $n$-omino exactly once in the grid. We present a new dynamic programming approach for this problem, allowing us to prove optimality for the pentomino and hexomino fence and to classify all possible pentomino orders by optimal area. For the first time we also provide strong heuristic solutions and upper bounds for all fences up to $n = 15$, after which we even approximate the $500$-omino fence with computational experiments. Finally, we generalize the problem to 3D, where we prove optimality for pentacubes and derive heuristic solutions for hexacubes and heptacubes.

Authors: Alexis Langlois-Rémillard, Mia N. Müßig, Helmut Podhaisky, Erika Roldan, Günter Rote, Maximilian Brömme, Matthes Norden, Maximilian Prietzel, Daniel Yu, Mykhailo Lyader

The $n$-omino fence problem asks how many cells can be surrounded by placing every $n$-omino exactly once in the grid. We present a new dynamic programming approach for this problem, allowing us to prove optimality for the pentomino and hexomino fence and to classify all possible pentomino orders by optimal area. For the first time we also provide strong heuristic solutions and upper bounds for all fences up to $n = 15$, after which we even approximate the $500$-omino fence with computational experiments. Finally, we generalize the problem to 3D, where we prove optimality for pentacubes and derive heuristic solutions for hexacubes and heptacubes.

Friday, November 13 2026, 01:00

An upper bound on the number of edges of a Gomory corner polyhedron

Authors: A. Yu. Chirkov, S. I. Veselov, D. V. Gribanov, D. S. Malyshev, N. Yu. Zolotykh

We establish the first nontrivial upper bound on the number of edges of a Gomory corner polyhedron, resolving a long-standing open question. For an $n$-dimensional corner polyhedron associated with an integer lattice of determinant $δ$, the number of edges is at most $$ O\left(n6^n \cdot \bigl(1+\log_2(2δ)\bigr)^{2n-3}\right). $$ This improves the standard bound obtained by counting pairs of vertices, reducing the logarithmic exponent from $(2n-2)$ to $(2n-3)$.

Authors: A. Yu. Chirkov, S. I. Veselov, D. V. Gribanov, D. S. Malyshev, N. Yu. Zolotykh

We establish the first nontrivial upper bound on the number of edges of a Gomory corner polyhedron, resolving a long-standing open question. For an $n$-dimensional corner polyhedron associated with an integer lattice of determinant $δ$, the number of edges is at most $$ O\left(n6^n \cdot \bigl(1+\log_2(2δ)\bigr)^{2n-3}\right). $$ This improves the standard bound obtained by counting pairs of vertices, reducing the logarithmic exponent from $(2n-2)$ to $(2n-3)$.

Friday, November 13 2026, 01:00

Visibility graphs of irregularly sampled two-dimensional fields via Voronoi tessellations

Authors: Francisco J. Muñoz, Juan Carlos Nuño

Visibility graphs map a time series into a network by linking two data whenever the straight line between them is not obstructed by intermediate data. Their extension to two-dimensional fields has so far been restricted to regular lattices, where visibility is checked along a finite number of lattice directions. We propose a definition for scalar data sampled at arbitrary, irregularly scattered points of the plane. Each point is assigned its Voronoi cell, so that the data become a landscape of prisms, and two points are linked when the line of sight between them clears the cells crossed by the connecting segment. This yields horizontal ($\HVGV$) and natural ($\NVGV$) Voronoi visibility graphs that reduce exactly to the classical horizontal and natural visibility graphs when the points are collinear, contain the Gabriel graph of the point set, and can be built by an exact walk through the Delaunay triangulation. For uncorrelated heights, the horizontal graph depends only on the ranks of the data, so its statistics are independent of the marginal distribution. The probability that two points see each other is $2/[(M+1)(M+2)]$, where $M$ is the number of Voronoi cells crossed, which reduces the problem to the geometry of the tessellation. For Poisson points this gives a degree distribution with a scale-free tail $P(k)\simeq(π^3/4)\,k^{-2}$ and a mean degree that grows as $(π^3/8)\ln N$, in contrast with the finite value $\langle k\rangle=4$ of one-dimensional series. Two dimensions are therefore the marginal case in which isotropic horizontal visibility ceases to be local. Numerical simulations with up to $N=1.28\times10^5$ points support these predictions.

Authors: Francisco J. Muñoz, Juan Carlos Nuño

Visibility graphs map a time series into a network by linking two data whenever the straight line between them is not obstructed by intermediate data. Their extension to two-dimensional fields has so far been restricted to regular lattices, where visibility is checked along a finite number of lattice directions. We propose a definition for scalar data sampled at arbitrary, irregularly scattered points of the plane. Each point is assigned its Voronoi cell, so that the data become a landscape of prisms, and two points are linked when the line of sight between them clears the cells crossed by the connecting segment. This yields horizontal ($\HVGV$) and natural ($\NVGV$) Voronoi visibility graphs that reduce exactly to the classical horizontal and natural visibility graphs when the points are collinear, contain the Gabriel graph of the point set, and can be built by an exact walk through the Delaunay triangulation. For uncorrelated heights, the horizontal graph depends only on the ranks of the data, so its statistics are independent of the marginal distribution. The probability that two points see each other is $2/[(M+1)(M+2)]$, where $M$ is the number of Voronoi cells crossed, which reduces the problem to the geometry of the tessellation. For Poisson points this gives a degree distribution with a scale-free tail $P(k)\simeq(π^3/4)\,k^{-2}$ and a mean degree that grows as $(π^3/8)\ln N$, in contrast with the finite value $\langle k\rangle=4$ of one-dimensional series. Two dimensions are therefore the marginal case in which isotropic horizontal visibility ceases to be local. Numerical simulations with up to $N=1.28\times10^5$ points support these predictions.

Friday, November 13 2026, 01:00

Tight Algorithms for Conservative Finite-Valued CSPs Parameterized by Treewidth

Authors: Jakob Greilhuber, Dániel Marx, George Osipov, Marta Piecyk, Paweł Rzążewski, Stanislav Živný

We consider valued constraint satisfaction problems (VCSPs) with a fixed finite domain $D$ and a fixed finite constraint language $Γ$, in the setting where $Γ$ contains all crisp unary constraints (called hard lists) and the remaining cost functions are finite-valued. We call such a language a finite-valued hard-list language. Let $c_Γ$ be the size of the largest set $S\subseteq D$ such that $Γ$ can express (using the usual VCSP closure operations) a binary disequality function on elements of $S$. Our main result tightly connects the complexity of the problem parameterized by treewidth to this value $c_Γ$. (1) The problem can be solved in time $(c_Γ)^{\mathsf{tw}} \cdot (|I| + |\mathcal{T}|)^{O(1)}$ when given an instance $I$ with a tree decomposition $\mathcal{T}$ of width $\mathsf{tw}$ of the primal graph of $I$. (2) If $c_Γ\geq 2$, there is no such algorithm solving the problem in time $(c_Γ- \varepsilon)^{\mathsf{tw}} \cdot (|I| + |\mathcal{T}|)^{O(1)}$ for any $\varepsilon > 0$, unless the Primal Treewidth SETH (and hence CNF-SETH) fails. In particular, this gives a far-reaching generalization of a similar result of Esmer et al. [ESA 2024] on the optimization version of list homomorphism for graphs.

Authors: Jakob Greilhuber, Dániel Marx, George Osipov, Marta Piecyk, Paweł Rzążewski, Stanislav Živný

We consider valued constraint satisfaction problems (VCSPs) with a fixed finite domain $D$ and a fixed finite constraint language $Γ$, in the setting where $Γ$ contains all crisp unary constraints (called hard lists) and the remaining cost functions are finite-valued. We call such a language a finite-valued hard-list language. Let $c_Γ$ be the size of the largest set $S\subseteq D$ such that $Γ$ can express (using the usual VCSP closure operations) a binary disequality function on elements of $S$. Our main result tightly connects the complexity of the problem parameterized by treewidth to this value $c_Γ$. (1) The problem can be solved in time $(c_Γ)^{\mathsf{tw}} \cdot (|I| + |\mathcal{T}|)^{O(1)}$ when given an instance $I$ with a tree decomposition $\mathcal{T}$ of width $\mathsf{tw}$ of the primal graph of $I$. (2) If $c_Γ\geq 2$, there is no such algorithm solving the problem in time $(c_Γ- \varepsilon)^{\mathsf{tw}} \cdot (|I| + |\mathcal{T}|)^{O(1)}$ for any $\varepsilon > 0$, unless the Primal Treewidth SETH (and hence CNF-SETH) fails. In particular, this gives a far-reaching generalization of a similar result of Esmer et al. [ESA 2024] on the optimization version of list homomorphism for graphs.

Friday, November 13 2026, 01:00

Individually Fair Facility Location

Authors: Rajni Dabas, Samir Khuller, Milind Maiti

We study individually fair facility location, where each client must connect to an open facility within a prescribed radius and the objective is to minimize total opening and connection cost. We show that preserving the radii changes the approximability of metric facility location fundamentally. For every fixed radius violation below three, no deterministic polynomial-time algorithm achieves a sublogarithmic cost factor unless $\mathrm{P}=\mathrm{NP}$. The hardness holds with uniform opening costs and uniform radii. It contradicts the strict-fairness constant-factor guarantee recently claimed by Liu, Xu, and Zhang [Theory of Computing Systems, Vol. 70 (2026)]. We then establish a positive result for arbitrary client radii. We give a randomized polynomial-time algorithm that has cost at most $1.725$ times the strictly fair optimum and serves every client within three times its fair radius (in the Appendix we show how this can be de-randomized). We use the approach of Byrka, Srinivasan and Swamy [IPCO 2010]. The facility-location radius factor three is therefore optimal among deterministic polynomial-time algorithms with constant cost factor, unless $\mathrm{P}=\mathrm{NP}$. Experiments on real-world data indicate that the worst case radius bound is rarely realized in practice: the algorithm serves over 99.8% of clients within their stated radius, and never beyond 1.5 times it. Our code is available at github.com/milindmaiti/IFFL_Implementation.

Authors: Rajni Dabas, Samir Khuller, Milind Maiti

We study individually fair facility location, where each client must connect to an open facility within a prescribed radius and the objective is to minimize total opening and connection cost. We show that preserving the radii changes the approximability of metric facility location fundamentally. For every fixed radius violation below three, no deterministic polynomial-time algorithm achieves a sublogarithmic cost factor unless $\mathrm{P}=\mathrm{NP}$. The hardness holds with uniform opening costs and uniform radii. It contradicts the strict-fairness constant-factor guarantee recently claimed by Liu, Xu, and Zhang [Theory of Computing Systems, Vol. 70 (2026)]. We then establish a positive result for arbitrary client radii. We give a randomized polynomial-time algorithm that has cost at most $1.725$ times the strictly fair optimum and serves every client within three times its fair radius (in the Appendix we show how this can be de-randomized). We use the approach of Byrka, Srinivasan and Swamy [IPCO 2010]. The facility-location radius factor three is therefore optimal among deterministic polynomial-time algorithms with constant cost factor, unless $\mathrm{P}=\mathrm{NP}$. Experiments on real-world data indicate that the worst case radius bound is rarely realized in practice: the algorithm serves over 99.8% of clients within their stated radius, and never beyond 1.5 times it. Our code is available at https://github.com/milindmaiti/IFFL_Implementation.

Friday, November 13 2026, 01:00

Sketching Language Models for Context Attribution

Authors: Christopher Liaw, Aranyak Mehta, Grigoris Velegkas, Felix Zhou

Context attribution asks how removing parts of a prompt changes a language model's output. We study leave-one-out (LOO) attribution for a prompt of $n$ tokens and a fixed measurement function. Computing all LOO values directly requires $n$ model evaluations. Under a stability assumption, our algorithm estimates all LOO values to error proportional to $\varepsilon$ with constant probability using $O(\sqrt{\log(1/\varepsilon)}/\varepsilon)$ evaluation calls. To complement our upper bound, we prove a matching $Ω(\sqrt{\log(1/\varepsilon)}/\varepsilon)$ lower bound on the number of evaluation calls in an oracle model. Empirically, we verify coefficient decay consistent with stability and evaluate the performance of our sketching algorithm on a small language-model retrieval task. Technically, our result is based on a sketching algorithm for arithmetic circuits whose Taylor coefficients satisfy the stability condition recently introduced by Gunn [Gun26]. A direct adaptation of their forward differentiation sketches would require $\widetilde{O}(\log(n)/\varepsilon^2)$ evaluation calls. Our key observation is that we can apply forward differentiation to the backpropagation circuit, with only constant overhead over a forward sketch of the same Taylor degree. This reduces the variance of the sketches and is key to our improved evaluation bound. Our lower bound builds on the techniques of Amsel, Chen, Duman Keles, Halikias, Musco, and Musco [ACDH+26, SIAM Journal on Matrix Analysis and Applications] for lower bounding the number of matrix-vector queries to estimate the diagonal elements of random matrices.

Authors: Christopher Liaw, Aranyak Mehta, Grigoris Velegkas, Felix Zhou

Context attribution asks how removing parts of a prompt changes a language model's output. We study leave-one-out (LOO) attribution for a prompt of $n$ tokens and a fixed measurement function. Computing all LOO values directly requires $n$ model evaluations. Under a stability assumption, our algorithm estimates all LOO values to error proportional to $\varepsilon$ with constant probability using $O(\sqrt{\log(1/\varepsilon)}/\varepsilon)$ evaluation calls. To complement our upper bound, we prove a matching $Ω(\sqrt{\log(1/\varepsilon)}/\varepsilon)$ lower bound on the number of evaluation calls in an oracle model. Empirically, we verify coefficient decay consistent with stability and evaluate the performance of our sketching algorithm on a small language-model retrieval task. Technically, our result is based on a sketching algorithm for arithmetic circuits whose Taylor coefficients satisfy the stability condition recently introduced by Gunn [Gun26]. A direct adaptation of their forward differentiation sketches would require $\widetilde{O}(\log(n)/\varepsilon^2)$ evaluation calls. Our key observation is that we can apply forward differentiation to the backpropagation circuit, with only constant overhead over a forward sketch of the same Taylor degree. This reduces the variance of the sketches and is key to our improved evaluation bound. Our lower bound builds on the techniques of Amsel, Chen, Duman Keles, Halikias, Musco, and Musco [ACDH+26, SIAM Journal on Matrix Analysis and Applications] for lower bounding the number of matrix-vector queries to estimate the diagonal elements of random matrices.

Friday, November 13 2026, 01:00

Integer Programming with Polyhedral Constraints on the Image

Authors: Anubhav Dhar, Tim Randolph, Karol Węgrzycki

We generalize integer programming to the setting in which the image $A \mathbf{x}$ is restricted to an explicitly given rational polyhedron $P$. That is, we solve $\min\{\mathbf{c}^{\mathsf T}\mathbf{x}:A\mathbf{x}\in P, \mathbf{x}\in\mathbb{Z}_{\geq 0}^n\}$, where $A\in\mathbb{Z}^{d\times n}$ and $P$ is an explicitly given rational polyhedron. We present a randomized algorithm running in time $(dΔ)^{O(d)}\operatorname{poly}(L)$ in this regime, where $Δ:= \|A\|_\infty$ and $L$ is the binary input length. This running time matches that of the best-known parameterized algorithms for integer programming, due to Eisenbrand and Weismantel and to Jansen and Rohwedder, and improves on the $O(d^2)$ exponent achieved by Dadush, Léonard, Rohwedder, and Verschae in a more general setting. This result enables several applications, including exact algorithms for integer programming and unbounded knapsack in which the consumed resources $A\mathbf{x}$ must satisfy arbitrary additional linear constraints. We also obtain a randomized fixed-parameter algorithm for Unbounded Subset Sum parameterized by the doubling constant, resolving an open problem of Randolph and Węgrzycki. For integer programs with $m$ constraints and bounded row doubling, we obtain an $f(C,m)n^{O(m)}\operatorname{poly}(L)$-time algorithm and show that the dependence on $m$ in the exponent of $n$ is necessary under $\mathsf{W[1]}\ne\mathsf{FPT}$.

Authors: Anubhav Dhar, Tim Randolph, Karol Węgrzycki

We generalize integer programming to the setting in which the image $A \mathbf{x}$ is restricted to an explicitly given rational polyhedron $P$. That is, we solve $\min\{\mathbf{c}^{\mathsf T}\mathbf{x}:A\mathbf{x}\in P, \mathbf{x}\in\mathbb{Z}_{\geq 0}^n\}$, where $A\in\mathbb{Z}^{d\times n}$ and $P$ is an explicitly given rational polyhedron. We present a randomized algorithm running in time $(dΔ)^{O(d)}\operatorname{poly}(L)$ in this regime, where $Δ:= \|A\|_\infty$ and $L$ is the binary input length. This running time matches that of the best-known parameterized algorithms for integer programming, due to Eisenbrand and Weismantel and to Jansen and Rohwedder, and improves on the $O(d^2)$ exponent achieved by Dadush, Léonard, Rohwedder, and Verschae in a more general setting. This result enables several applications, including exact algorithms for integer programming and unbounded knapsack in which the consumed resources $A\mathbf{x}$ must satisfy arbitrary additional linear constraints. We also obtain a randomized fixed-parameter algorithm for Unbounded Subset Sum parameterized by the doubling constant, resolving an open problem of Randolph and Węgrzycki. For integer programs with $m$ constraints and bounded row doubling, we obtain an $f(C,m)n^{O(m)}\operatorname{poly}(L)$-time algorithm and show that the dependence on $m$ in the exponent of $n$ is necessary under $\mathsf{W[1]}\ne\mathsf{FPT}$.

Friday, November 13 2026, 01:00

Separations Between Submodular-Like Graph Widths

Authors: Karl Bringmann, Egor Gorbachev

Many algorithmic frameworks exist for evaluating join queries. They range from simple ones, such as dynamic programming over a single tree decomposition, high-degree-low-degree splitting from graph algorithms, and the degree uniformization of Joglekar and Ré, to sophisticated ones based on information-theoretic inequalities, such as the PANDA framework for submodular width. Each of these frameworks except high-degree-low-degree splitting comes with a width notion of the query that determines its running-time exponent. We fill this gap by formalizing high-degree-low-degree splitting with $\ell$ degree classes, which gives rise to the degree-splitting width $\mathrm{ds}_\ell$-width. These width notions form a hierarchy, but it was not known whether the more complicated frameworks achieve better worst-case exponents than the simpler ones on any graph query. We study this question for full self-join-free join queries with binary relations, which amounts to colored subgraph isomorphism for a pattern graph $H$. We show that every inequality in the hierarchy $\mathrm{subw} \le \#\mathrm{subw} \le \text{m-width} \le \mathrm{ds}_\ell\text{-width}$ can be strict. For instance, $\text{m-width}(K_{3,3}) = 2 < \mathrm{ds}_\ell\text{-width}(K_{3,3})$ for every constant $\ell$, so no constant number of degree classes reaches the exponent of degree uniformization. We find the smallest separating patterns, with 8, 9, and 6 nodes, respectively, by an exhaustive search with a software tool that computes these widths via mixed integer linear programming. Along the way, we give a degree-based algorithm that removes the polylogarithmic overhead of degree uniformization when evaluating full join queries. Furthermore, we show that with logarithmically many degree classes, the degree-splitting width coincides with the m-width of Joglekar and Ré. Finally, we discuss several open problems.

Authors: Karl Bringmann, Egor Gorbachev

Many algorithmic frameworks exist for evaluating join queries. They range from simple ones, such as dynamic programming over a single tree decomposition, high-degree-low-degree splitting from graph algorithms, and the degree uniformization of Joglekar and Ré, to sophisticated ones based on information-theoretic inequalities, such as the PANDA framework for submodular width. Each of these frameworks except high-degree-low-degree splitting comes with a width notion of the query that determines its running-time exponent. We fill this gap by formalizing high-degree-low-degree splitting with $\ell$ degree classes, which gives rise to the degree-splitting width $\mathrm{ds}_\ell$-width. These width notions form a hierarchy, but it was not known whether the more complicated frameworks achieve better worst-case exponents than the simpler ones on any graph query. We study this question for full self-join-free join queries with binary relations, which amounts to colored subgraph isomorphism for a pattern graph $H$. We show that every inequality in the hierarchy $\mathrm{subw} \le \#\mathrm{subw} \le \text{m-width} \le \mathrm{ds}_\ell\text{-width}$ can be strict. For instance, $\text{m-width}(K_{3,3}) = 2 < \mathrm{ds}_\ell\text{-width}(K_{3,3})$ for every constant $\ell$, so no constant number of degree classes reaches the exponent of degree uniformization. We find the smallest separating patterns, with 8, 9, and 6 nodes, respectively, by an exhaustive search with a software tool that computes these widths via mixed integer linear programming. Along the way, we give a degree-based algorithm that removes the polylogarithmic overhead of degree uniformization when evaluating full join queries. Furthermore, we show that with logarithmically many degree classes, the degree-splitting width coincides with the m-width of Joglekar and Ré. Finally, we discuss several open problems.

Friday, November 13 2026, 01:00

Dynamic $k$-Median with Worst-Case Update Time and Recourse

Authors: Chenglin Fan

We study fully dynamic metric $k$-median under unit-weight client insertions and deletions. After each update, the algorithm must maintain at most $k$ centers chosen from the current client locations. We give a randomized constant-factor approximation algorithm with $\widetilde O(k)$ worst-case update time and $O((1+\logΔ)^3)$ worst-case center recourse, where $Δ$ is the aspect ratio of the underlying metric space. The approximation guarantee holds simultaneously over any prescribed finite adaptive update sequence with high probability, while the update time and recourse bounds hold on every execution. We also obtain the same asymptotic guarantees for metric $k$-means.

Authors: Chenglin Fan

We study fully dynamic metric $k$-median under unit-weight client insertions and deletions. After each update, the algorithm must maintain at most $k$ centers chosen from the current client locations. We give a randomized constant-factor approximation algorithm with $\widetilde O(k)$ worst-case update time and $O((1+\logΔ)^3)$ worst-case center recourse, where $Δ$ is the aspect ratio of the underlying metric space. The approximation guarantee holds simultaneously over any prescribed finite adaptive update sequence with high probability, while the update time and recourse bounds hold on every execution. We also obtain the same asymptotic guarantees for metric $k$-means.

Friday, November 13 2026, 01:00

The Submodular Secretary Problem: Tight $1-1/e$ for Monotone Objectives, and Strictly Between $1/e$ and $1/2$ in General

Authors: Hamed Abdi, Kiarash Banihashem, MohammadTaghi Hajiaghayi, Danny Mittal

In the submodular secretary problem with a cardinality constraint, $n$ elements arrive in uniformly random order, and an algorithm must irrevocably accept or reject each one on arrival so as to maximize a nonnegative submodular function over sets of at most $k$ elements. For monotone objectives we settle the problem: the optimal competitive ratio tends to exactly $1-1/e$ as $k\to\infty$, and its worst case over all budgets $k$ is exactly $1/e$. The bound $1-1/e$ cannot be exceeded even with unlimited computation, and it is achieved by a single algorithm with at most $4n+1$ value queries; previously the best ratio without a shortlist was $1/e-o(1)$, with exponentially many queries. For general objectives, where the best known ratio was about $0.107$, we show that the optimal limiting ratio lies strictly between $1/e$ and $1/2$: we achieve $1/e+3.5\cdot10^{-6}$ for large $k$, and no algorithm achieves more than $0.497053$. We also achieve $0.2677$ at every budget. The main new ingredient of our algorithm is eligibility attenuation: an element that passes the price test at time $t$ is accepted with probability $e^{-τ}$, where $τ$ is the total earlier time during which it would have passed, which makes the probability that it has already been selected exactly $1-e^{-τ}$.

Authors: Hamed Abdi, Kiarash Banihashem, MohammadTaghi Hajiaghayi, Danny Mittal

In the submodular secretary problem with a cardinality constraint, $n$ elements arrive in uniformly random order, and an algorithm must irrevocably accept or reject each one on arrival so as to maximize a nonnegative submodular function over sets of at most $k$ elements. For monotone objectives we settle the problem: the optimal competitive ratio tends to exactly $1-1/e$ as $k\to\infty$, and its worst case over all budgets $k$ is exactly $1/e$. The bound $1-1/e$ cannot be exceeded even with unlimited computation, and it is achieved by a single algorithm with at most $4n+1$ value queries; previously the best ratio without a shortlist was $1/e-o(1)$, with exponentially many queries. For general objectives, where the best known ratio was about $0.107$, we show that the optimal limiting ratio lies strictly between $1/e$ and $1/2$: we achieve $1/e+3.5\cdot10^{-6}$ for large $k$, and no algorithm achieves more than $0.497053$. We also achieve $0.2677$ at every budget. The main new ingredient of our algorithm is eligibility attenuation: an element that passes the price test at time $t$ is accepted with probability $e^{-τ}$, where $τ$ is the total earlier time during which it would have passed, which makes the probability that it has already been selected exactly $1-e^{-τ}$.

Friday, November 13 2026, 01:00

Matching in Random-Order Streams via Historical Deletion Bounds

Authors: Alireza Farhadi, Mohammadtaghi Hajiaghayi, Mohammad Mahdavi

We study maximum matching when the edges of a fixed graph arrive in a uniformly random order. For any fixed $\varepsilon>0$, we give a one-pass algorithm that achieves, with high probability over the arrival order, approximation ratios of $0.71295$ for unweighted graphs and $0.71295-\varepsilon$ for weighted graphs, using near-linear space with an explicit charge for weight precision. Both approximation ratios hold for bipartite and general graphs. The algorithm does not require the stream length or the weight range in advance. The main structural ingredient is a historical deletion inequality for weighted matchings. The deletion price of a vertex set is the decrease in maximum matching weight when that set is removed. We bound the sum of these prices for pairwise disjoint vertex sets, even when each price is evaluated at a different stage as edges are added to the graph. Applied to the usual unfolding of integer weights, it transfers a bipartite cardinality certificate to an original weighted matching, even when the original graph is not bipartite. This transfer preserves the approximation coefficient. We combine this transfer with a refined analysis of fractional matchings in the two auxiliary graphs arising in the transfer theorem. An exact rational linear-programming certificate proves the stated constant for every feasible graph profile. The approximation guarantees and the new structural results are formally verified in Lean.

Authors: Alireza Farhadi, Mohammadtaghi Hajiaghayi, Mohammad Mahdavi

We study maximum matching when the edges of a fixed graph arrive in a uniformly random order. For any fixed $\varepsilon>0$, we give a one-pass algorithm that achieves, with high probability over the arrival order, approximation ratios of $0.71295$ for unweighted graphs and $0.71295-\varepsilon$ for weighted graphs, using near-linear space with an explicit charge for weight precision. Both approximation ratios hold for bipartite and general graphs. The algorithm does not require the stream length or the weight range in advance. The main structural ingredient is a historical deletion inequality for weighted matchings. The deletion price of a vertex set is the decrease in maximum matching weight when that set is removed. We bound the sum of these prices for pairwise disjoint vertex sets, even when each price is evaluated at a different stage as edges are added to the graph. Applied to the usual unfolding of integer weights, it transfers a bipartite cardinality certificate to an original weighted matching, even when the original graph is not bipartite. This transfer preserves the approximation coefficient. We combine this transfer with a refined analysis of fractional matchings in the two auxiliary graphs arising in the transfer theorem. An exact rational linear-programming certificate proves the stated constant for every feasible graph profile. The approximation guarantees and the new structural results are formally verified in Lean.

Friday, November 13 2026, 01:00

Linear Regression with Self-Selection Bias in Polynomial Time

Authors: Felix Zhou

We revisit the problem of estimating $k$ linear regressors with self-selection bias in $n$ dimensions up to error $\varepsilon$ with the maximum self-selection criterion, as introduced by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [CDIZ23, STOC]. Our main result is a $\mathrm{poly}(n,k,1/\varepsilon)$ time algorithm for the unknown-index variant of this problem, thereby improving upon the running times of the best known algorithms due to Gaitonde and Mossel [GM24] and Kalavasis, Mehrotra, and Zhou [KMZ26, COLT] with exponential dependence on $k$. Our algorithm succeeds in the presence of correlated noise between regressors, which captures the related setting of max-affine regression. Consequently, we also give the first polynomial-time algorithm for this problem, improving on the prior works of Ghosh, Pananjady, Guntuboyina, and Ramchandran [GPGR22, IEEE Transactions on Information Theory] and Gaitonde and Mossel [GM24] with exponential dependence on $k$. Prior works exhaustively search over a $k$-dimensional subspace spanned by the regressors, using conditional moments of the response to test candidates. In contrast, our algorithm is driven by an improved understanding of the moments of the covariates. We show that, restricting to observations with large response values, the degree-$(\log n)$ Hermite moment of the covariates approximately decomposes as a sum of tensor powers of the regressors. The first step of our algorithm can be viewed as a tensor decomposition algorithm for these implicitly represented high-order tensors, which outputs unit vectors parallel to the regressors. A secondary step based on a simple variant of ordinary least squares extracts estimates for the regressors given these unit vectors. We believe our analytic insight and algorithmic components may have applications to designing efficient algorithms for other latent-variable problems.

Authors: Felix Zhou

We revisit the problem of estimating $k$ linear regressors with self-selection bias in $n$ dimensions up to error $\varepsilon$ with the maximum self-selection criterion, as introduced by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [CDIZ23, STOC]. Our main result is a $\mathrm{poly}(n,k,1/\varepsilon)$ time algorithm for the unknown-index variant of this problem, thereby improving upon the running times of the best known algorithms due to Gaitonde and Mossel [GM24] and Kalavasis, Mehrotra, and Zhou [KMZ26, COLT] with exponential dependence on $k$. Our algorithm succeeds in the presence of correlated noise between regressors, which captures the related setting of max-affine regression. Consequently, we also give the first polynomial-time algorithm for this problem, improving on the prior works of Ghosh, Pananjady, Guntuboyina, and Ramchandran [GPGR22, IEEE Transactions on Information Theory] and Gaitonde and Mossel [GM24] with exponential dependence on $k$. Prior works exhaustively search over a $k$-dimensional subspace spanned by the regressors, using conditional moments of the response to test candidates. In contrast, our algorithm is driven by an improved understanding of the moments of the covariates. We show that, restricting to observations with large response values, the degree-$(\log n)$ Hermite moment of the covariates approximately decomposes as a sum of tensor powers of the regressors. The first step of our algorithm can be viewed as a tensor decomposition algorithm for these implicitly represented high-order tensors, which outputs unit vectors parallel to the regressors. A secondary step based on a simple variant of ordinary least squares extracts estimates for the regressors given these unit vectors. We believe our analytic insight and algorithmic components may have applications to designing efficient algorithms for other latent-variable problems.

Friday, November 13 2026, 01:00

Combinatorial Bernoulli Factories with Polynomially Many Coins

Authors: Renato Paes Leme, Rad Niazadeh

We provide an efficient implementation of combinatorial Bernoulli factories. Such a factory turns coins of unknown biases $x$ into a random vertex of a polytope $P$ whose mean is exactly $x$. For every affine slice of the cube (the class of polytopes that admit a factory), we give a single algorithm whose expected number of coin flips and running time at every interior $x$ are polynomial in the dimension and in $1/{δ_*}$, where ${δ_*}=\min_i\{x_i,1-x_i\}$ is never supplied to the algorithm. For perfect matchings this gives $O(n^{8}{δ_*}^{-2}\log(n/{δ_*}))$ flips, while the factory of Niazadeh, Paes Leme, and Schneider [STOC 2021] spends at least $n^{2n-1}$ flips in expectation at the uniform doubly stochastic matrix. For other polytopes, our method rounds exactly given a point deep inside $P$ and the promise that $x$ is far from the boundary of $P$.

Authors: Renato Paes Leme, Rad Niazadeh

We provide an efficient implementation of combinatorial Bernoulli factories. Such a factory turns coins of unknown biases $x$ into a random vertex of a polytope $P$ whose mean is exactly $x$. For every affine slice of the cube (the class of polytopes that admit a factory), we give a single algorithm whose expected number of coin flips and running time at every interior $x$ are polynomial in the dimension and in $1/{δ_*}$, where ${δ_*}=\min_i\{x_i,1-x_i\}$ is never supplied to the algorithm. For perfect matchings this gives $O(n^{8}{δ_*}^{-2}\log(n/{δ_*}))$ flips, while the factory of Niazadeh, Paes Leme, and Schneider [STOC 2021] spends at least $n^{2n-1}$ flips in expectation at the uniform doubly stochastic matrix. For other polytopes, our method rounds exactly given a point deep inside $P$ and the promise that $x$ is far from the boundary of $P$.

Friday, November 13 2026, 01:00

Faster Deterministic All-Pairs Shortest Paths with Real Edge Weights

Authors: Isaac Barouch Essayag, Aryeh Lev Zabokritskiy

For directed $n$-vertex graphs with arbitrary finite real edge weights and no negative cycles, we compute all-pairs distances deterministically in $O(n^{2.9929729})$ worst-case time using Uznański's half-width matrix multiplication bound. Using only the Le Gall--Urrutia rectangular multiplication bounds, our algorithm runs in $O(n^{2.99306273})$ time. This improves both the deterministic bound of $O(n^{2.9997})$ due to Chan, Itakura, Ma and Yu and the Las Vegas expected-time bound of $O(n^{2.993805})$ reported in the shreshth-s working note.

Authors: Isaac Barouch Essayag, Aryeh Lev Zabokritskiy

For directed $n$-vertex graphs with arbitrary finite real edge weights and no negative cycles, we compute all-pairs distances deterministically in $O(n^{2.9929729})$ worst-case time using Uznański's half-width matrix multiplication bound. Using only the Le Gall--Urrutia rectangular multiplication bounds, our algorithm runs in $O(n^{2.99306273})$ time. This improves both the deterministic bound of $O(n^{2.9997})$ due to Chan, Itakura, Ma and Yu and the Las Vegas expected-time bound of $O(n^{2.993805})$ reported in the shreshth-s working note.

Friday, November 13 2026, 01:00

A Little Length Slack Goes a Long Way in Planar Length-Constrained Network Design

Authors: D Ellis Hershkowitz, Richard Z Huang, Rhea Jain, Manas Korimilli

Problems in length-constrained network design ask for min-cost subgraphs connecting $k$ input vertex pairs by paths of length at most $h$. Algorithms that instead connect these pairs by paths of length at most $βh$ are said to have length slack $β$. Recent work has shown that poly-time polylogarithmic approximations with polylogarithmic length slack are possible for a broad class of both directed and multicommodity length-constrained network design problems on planar graphs. We give the first poly-time polylogarithmic approximations with $O(1)$ length slack for these problems on planar graphs. In particular, for the length-constrained setting, we give LP-competitive algorithms for planar graphs with length slack $1+ε$ and approximation ratios of $O(\log^2 k/ε)$ for directed Steiner tree and $O(\log^5 k/ε^2)$ for (undirected) Steiner forest. Similarly, for length-constrained directed Steiner forest, we give an $O(\log^6 k/ε)$ approximation on planar graphs with length slack $6+ε$. Our approach is based on a novel charging argument that charges the cost of our solution to the length of the LP and the length of our solution to the cost of the LP. We complement these algorithms with hardness results that separate planar and general graphs for both forest problems: under standard complexity assumptions, we rule out polylogarithmic approximations on general graphs with length slack below $3$.

Authors: D Ellis Hershkowitz, Richard Z Huang, Rhea Jain, Manas Korimilli

Problems in length-constrained network design ask for min-cost subgraphs connecting $k$ input vertex pairs by paths of length at most $h$. Algorithms that instead connect these pairs by paths of length at most $βh$ are said to have length slack $β$. Recent work has shown that poly-time polylogarithmic approximations with polylogarithmic length slack are possible for a broad class of both directed and multicommodity length-constrained network design problems on planar graphs. We give the first poly-time polylogarithmic approximations with $O(1)$ length slack for these problems on planar graphs. In particular, for the length-constrained setting, we give LP-competitive algorithms for planar graphs with length slack $1+ε$ and approximation ratios of $O(\log^2 k/ε)$ for directed Steiner tree and $O(\log^5 k/ε^2)$ for (undirected) Steiner forest. Similarly, for length-constrained directed Steiner forest, we give an $O(\log^6 k/ε)$ approximation on planar graphs with length slack $6+ε$. Our approach is based on a novel charging argument that charges the cost of our solution to the length of the LP and the length of our solution to the cost of the LP. We complement these algorithms with hardness results that separate planar and general graphs for both forest problems: under standard complexity assumptions, we rule out polylogarithmic approximations on general graphs with length slack below $3$.

Friday, November 13 2026, 01:00

Near-Optimal Regret for Oracle-Efficient Adversarial Contextual Bandits

Authors: Kiarash Banihashem, MohammadTaghi Hajiaghayi

We study contextual bandits with $K$ actions, a class of $N$ policies, and i.i.d.\ contexts from a distribution that the learner can sample. For any sequence of loss functions fixed before play, we give an oracle-efficient algorithm with expected regret $O(\sqrt{KT\log(2N)}\,\log(2KT))$ against the best policy in hindsight on the realized contexts. This improves the $T^{2/3}$ dependence on the horizon in prior oracle-efficient bounds and achieves the optimal regret rate in this setting up to logarithmic factors. Our algorithm perturbs policy scores using a Poisson sample of auxiliary contexts. At each sampled context, the perturbation assigned to an action is a Laplace noise draw multiplied by a scale specific to that context and action. We show that choosing these scales to minimize a convex objective determined by past observations yields the stated regret bound. We further approximate the minimizing scales by projected stochastic gradient descent using only an exact offline cost-sensitive optimization oracle that returns the optimum value. Since the table of scales can be large, the algorithm represents the SGD iterates implicitly by storing the random draws that determine each update and evaluates the scales only at queried contexts. Both the oracle complexity and the remaining resource bounds are polynomial in $T$, $K$, and $\log(2N)$.

Authors: Kiarash Banihashem, MohammadTaghi Hajiaghayi

We study contextual bandits with $K$ actions, a class of $N$ policies, and i.i.d.\ contexts from a distribution that the learner can sample. For any sequence of loss functions fixed before play, we give an oracle-efficient algorithm with expected regret $O(\sqrt{KT\log(2N)}\,\log(2KT))$ against the best policy in hindsight on the realized contexts. This improves the $T^{2/3}$ dependence on the horizon in prior oracle-efficient bounds and achieves the optimal regret rate in this setting up to logarithmic factors. Our algorithm perturbs policy scores using a Poisson sample of auxiliary contexts. At each sampled context, the perturbation assigned to an action is a Laplace noise draw multiplied by a scale specific to that context and action. We show that choosing these scales to minimize a convex objective determined by past observations yields the stated regret bound. We further approximate the minimizing scales by projected stochastic gradient descent using only an exact offline cost-sensitive optimization oracle that returns the optimum value. Since the table of scales can be large, the algorithm represents the SGD iterates implicitly by storing the random draws that determine each update and evaluates the scales only at queried contexts. Both the oracle complexity and the remaining resource bounds are polynomial in $T$, $K$, and $\log(2N)$.

Friday, November 13 2026, 01:00

Nearly Optimal Dilation for Non-Uniform k-Center with Constant-Factor Budget Violation

Authors: Jannis Blauth, Siri Prüßner

In Non-Uniform $k$-Center, we are given a metric space $(V,d)$ and $L$ levels of balls with prescribed radii $r_1 \ge \ldots \ge r_L$ and nonnegative integer budgets $k_1,\ldots,k_L$. The goal is to place at most $k_\ell$ centers of each level $\ell \in [L]$ in $V$ and minimize the dilation $λ$ such that the corresponding balls of radii $λr_1,\dots,λr_L$ cover $V$. While this problem admits no subexponential approximation factor unless P=NP, the picture changes drastically when additional centers are allowed. An $(α,β)$-bicriteria approximation to Smooth Non-Uniform $k$-Center covers $V$ with dilation $βλ_{\text{opt}}$ such that, for each $\ell \in [L]$, the number of centers of the first $\ell$ levels is at most $α$ times their total budget. For every fixed $\varepsilon>0$, we give a randomized $(1+\varepsilon,8+\varepsilon)$-approximation and a deterministic $(O_{\varepsilon}(1),2+\varepsilon)$-approximation, both running in polynomial time. Even for classical $k$-Center, obtaining an $(O(1),2-δ)$-approximation is NP-hard for every fixed $δ\in (0,1]$. Thus, our second result almost settles the optimal dilation when a constant-factor violation of the budget is allowed, unless P=NP. We combine a new primal-dual enumeration procedure with a simple bucketing argument that yields a new trade-off between the two approximation factors. It remains open whether there is a polynomial-time $(1+\varepsilon,2+\varepsilon)$-bicriteria approximation, which would almost match the classical $2$-approximation for $k$-Center.

Authors: Jannis Blauth, Siri Prüßner

In Non-Uniform $k$-Center, we are given a metric space $(V,d)$ and $L$ levels of balls with prescribed radii $r_1 \ge \ldots \ge r_L$ and nonnegative integer budgets $k_1,\ldots,k_L$. The goal is to place at most $k_\ell$ centers of each level $\ell \in [L]$ in $V$ and minimize the dilation $λ$ such that the corresponding balls of radii $λr_1,\dots,λr_L$ cover $V$. While this problem admits no subexponential approximation factor unless P=NP, the picture changes drastically when additional centers are allowed. An $(α,β)$-bicriteria approximation to Smooth Non-Uniform $k$-Center covers $V$ with dilation $βλ_{\text{opt}}$ such that, for each $\ell \in [L]$, the number of centers of the first $\ell$ levels is at most $α$ times their total budget. For every fixed $\varepsilon>0$, we give a randomized $(1+\varepsilon,8+\varepsilon)$-approximation and a deterministic $(O_{\varepsilon}(1),2+\varepsilon)$-approximation, both running in polynomial time. Even for classical $k$-Center, obtaining an $(O(1),2-δ)$-approximation is NP-hard for every fixed $δ\in (0,1]$. Thus, our second result almost settles the optimal dilation when a constant-factor violation of the budget is allowed, unless P=NP. We combine a new primal-dual enumeration procedure with a simple bucketing argument that yields a new trade-off between the two approximation factors. It remains open whether there is a polynomial-time $(1+\varepsilon,2+\varepsilon)$-bicriteria approximation, which would almost match the classical $2$-approximation for $k$-Center.

Friday, November 13 2026, 01:00

Online Transportation via Shifted Geometric Scales

Authors: John Jeang

In the Online Transportation problem, introduced by Kalyanasundaram and Pruhs (1995), requests arrive sequentially in a metric space and must be assigned immediately and irrevocably to one of $m$ facilities with available capacity. The objective is to minimize the total distance between requests and their assigned facilities. The problem generalizes the Online Metric Matching problem, in which each facility has unit capacity, and inherits a $2m-1$ lower bound on the competitive ratio for all deterministic algorithms. For Online Metric Matching, Kalyanasundaram and Pruhs (1993) and Khuller, Mitchell, and Vazirani (1994) established an optimal deterministic competitive ratio of $2m-1$, attained by the Retrospective (Permutation) algorithm. Kalyanasundaram and Pruhs (1998) conjectured that the same ratio is achievable for online transportation. Recently, Seth, Das, and Raghvendra (2026) obtained a competitive ratio of approximately $6.66m-2.89$ using their Robustified Greedy algorithm. We give an algorithm with competitive ratio $ρ(m-1)+1$, where $ρ\approx5.64$. Our algorithm builds on minimum-spanning-tree search techniques developed by Harada and Itoh (2025) and Arndt, Moseley, Pruhs, and Uetz (2026), using priority-based virtual depth-first searches over the known facilities. However our algorithm uses a phase to shift the geometric search scales, to decide when to skip certain searches and optimize the thresholds that define what edges to include in a given search.

Authors: John Jeang

In the Online Transportation problem, introduced by Kalyanasundaram and Pruhs (1995), requests arrive sequentially in a metric space and must be assigned immediately and irrevocably to one of $m$ facilities with available capacity. The objective is to minimize the total distance between requests and their assigned facilities. The problem generalizes the Online Metric Matching problem, in which each facility has unit capacity, and inherits a $2m-1$ lower bound on the competitive ratio for all deterministic algorithms. For Online Metric Matching, Kalyanasundaram and Pruhs (1993) and Khuller, Mitchell, and Vazirani (1994) established an optimal deterministic competitive ratio of $2m-1$, attained by the Retrospective (Permutation) algorithm. Kalyanasundaram and Pruhs (1998) conjectured that the same ratio is achievable for online transportation. Recently, Seth, Das, and Raghvendra (2026) obtained a competitive ratio of approximately $6.66m-2.89$ using their Robustified Greedy algorithm. We give an algorithm with competitive ratio $ρ(m-1)+1$, where $ρ\approx5.64$. Our algorithm builds on minimum-spanning-tree search techniques developed by Harada and Itoh (2025) and Arndt, Moseley, Pruhs, and Uetz (2026), using priority-based virtual depth-first searches over the known facilities. However our algorithm uses a phase to shift the geometric search scales, to decide when to skip certain searches and optimize the thresholds that define what edges to include in a given search.

Friday, November 13 2026, 01:00

List Update with Time Windows and Delays without Batch-Serving

Authors: Icey Siyi Ai, Jingwei Li, Clifford Stein

We study list update with time windows and delays when every request requires a separate access. Unlike batch-serving models, serving several requests to one element still incurs one access per request, and each access may change the list for subsequent requests. For time windows of overlap at most $k\ge2$, our deterministic First-Element Removal algorithm is $(3-2/k)$-competitive, hence $3$-competitive in general. Against an oblivious adversary, a randomized adaptation of BIT is $7/4$-competitive on arbitrary $2$-overlap inputs. More generally, on inputs without self-overlap, it is $\left(13/4-9/[2(k+1)]\right)$-competitive for every $k\ge2$. For delays, we give a deterministic $5$-competitive algorithm and a lower bound of $2+\sqrt3$ for deterministic delay-unaware algorithms. Our analyses extend inversion potentials with per-request credits that reconcile different service times and orders.

Authors: Icey Siyi Ai, Jingwei Li, Clifford Stein

We study list update with time windows and delays when every request requires a separate access. Unlike batch-serving models, serving several requests to one element still incurs one access per request, and each access may change the list for subsequent requests. For time windows of overlap at most $k\ge2$, our deterministic First-Element Removal algorithm is $(3-2/k)$-competitive, hence $3$-competitive in general. Against an oblivious adversary, a randomized adaptation of BIT is $7/4$-competitive on arbitrary $2$-overlap inputs. More generally, on inputs without self-overlap, it is $\left(13/4-9/[2(k+1)]\right)$-competitive for every $k\ge2$. For delays, we give a deterministic $5$-competitive algorithm and a lower bound of $2+\sqrt3$ for deterministic delay-unaware algorithms. Our analyses extend inversion potentials with per-request credits that reconcile different service times and orders.

Friday, November 13 2026, 01:00

Finding the Farthest in High Dimensions: Dynamic Diameter and Farthest-Neighbor Search

Authors: Mohammad Hajiaghayi, Morteza Monemizadeh

We give faster algorithms for diameter and farthest-neighbor search in high-dimensional Euclidean space. For every fixed $1

Authors: Mohammad Hajiaghayi, Morteza Monemizadeh

We give faster algorithms for diameter and farthest-neighbor search in high-dimensional Euclidean space. For every fixed $1

Friday, November 13 2026, 01:00

A PTAS for Constraint Satisfaction Problems on Interval Graphs and Bounded Rank-Width Graphs

Authors: Ian DeHaan, Euiwoong Lee

We present a polynomial-time approximation scheme (PTAS) for all Boolean arity-two maximum constraint satisfaction problems (Boolean Max-2-CSP) where the constraint graph is either interval or has bounded rank-width. This class of problems includes, for example, Max-Cut, Max-DiCut, Max-2-Lin(2), and Max-2-SAT. In particular, this settles the approximability of Max-Cut on interval graphs. To achieve these results, we show the following. 1. For any $\varepsilon > 0$, there is an $n^{\mathcal O(r \varepsilon^{-2})}$-time $(1 - \varepsilon$)-approximation for Boolean Max-2-CSP on graphs of rank-width at most $r$. 2. There is a polynomial-time algorithm that, for any interval graph $G = (V, E)$ and $\varepsilon > 0$, finds a subgraph $H = (V, F)$ of $G$ such that $|F| \geq (1 - \varepsilon) |E|$ and $H$ has rank-width at most $\operatorname{poly}(\varepsilon^{-1})$. Our techniques additionally extend to all bounded-alphabet arity-two max CSPs.

Authors: Ian DeHaan, Euiwoong Lee

We present a polynomial-time approximation scheme (PTAS) for all Boolean arity-two maximum constraint satisfaction problems (Boolean Max-2-CSP) where the constraint graph is either interval or has bounded rank-width. This class of problems includes, for example, Max-Cut, Max-DiCut, Max-2-Lin(2), and Max-2-SAT. In particular, this settles the approximability of Max-Cut on interval graphs. To achieve these results, we show the following. 1. For any $\varepsilon > 0$, there is an $n^{\mathcal O(r \varepsilon^{-2})}$-time $(1 - \varepsilon$)-approximation for Boolean Max-2-CSP on graphs of rank-width at most $r$. 2. There is a polynomial-time algorithm that, for any interval graph $G = (V, E)$ and $\varepsilon > 0$, finds a subgraph $H = (V, F)$ of $G$ such that $|F| \geq (1 - \varepsilon) |E|$ and $H$ has rank-width at most $\operatorname{poly}(\varepsilon^{-1})$. Our techniques additionally extend to all bounded-alphabet arity-two max CSPs.

Friday, November 13 2026, 01:00

Edge-wise Graph Alignments

Authors: Akbar Davoodi, Christoph Flamm, Daniel Merkle, Peter F. Stadler

Pairwise graph ``alignments'', usually defined as maximum common subgraphs, have been considered in many application domains, including the comparison of molecular structures. Most commonly, the problem is formulated as maximum common induced subgraph (MCIS) or maximum common edge subgraph (MCES) problems. Sometimes these formulations are restricted to connected common subgraphs. Common subgraph problems readily generalize to more than two input graphs. However, this concept is fundamentally distinct from the notion of multiple sequence alignments. Abstractly, alignments are defined as super-structures from which the individual input objects are recovered via structure-preserving projections. While the mathematical construction readily applies to common induced subgraphs, it does not fully encompass the common edge subgraph problem, which is the variant more commonly employed in cheminformatics applications. In order to consistently define multiple edge-based alignments of graphs we represent a graph $G$ as E-graph $(E,\mathscr{V})$, where $E$ is the edge set of $G$ and $\mathscr{V}$ is the set of sets of incident edges. This representation implicitly encodes the vertices in terms of edge data. Alignments of E-graphs are themselves E-graphs that contain the input (E-)graphs as subgraphs. Maximum edge subgraphs correspond to maximum induced sub-E-graphs, and they can thus be computed using a slightly modified version of the VF2 algorithm without the need to switch from graphs to line graphs. Furthermore, multiple alignments of (E-)graphs yield a natural definition of a consensus graph that is ``on average'' present in the input without having to be a subgraph of every (and in fact of any) input graph.

Authors: Akbar Davoodi, Christoph Flamm, Daniel Merkle, Peter F. Stadler

Pairwise graph ``alignments'', usually defined as maximum common subgraphs, have been considered in many application domains, including the comparison of molecular structures. Most commonly, the problem is formulated as maximum common induced subgraph (MCIS) or maximum common edge subgraph (MCES) problems. Sometimes these formulations are restricted to connected common subgraphs. Common subgraph problems readily generalize to more than two input graphs. However, this concept is fundamentally distinct from the notion of multiple sequence alignments. Abstractly, alignments are defined as super-structures from which the individual input objects are recovered via structure-preserving projections. While the mathematical construction readily applies to common induced subgraphs, it does not fully encompass the common edge subgraph problem, which is the variant more commonly employed in cheminformatics applications. In order to consistently define multiple edge-based alignments of graphs we represent a graph $G$ as E-graph $(E,\mathscr{V})$, where $E$ is the edge set of $G$ and $\mathscr{V}$ is the set of sets of incident edges. This representation implicitly encodes the vertices in terms of edge data. Alignments of E-graphs are themselves E-graphs that contain the input (E-)graphs as subgraphs. Maximum edge subgraphs correspond to maximum induced sub-E-graphs, and they can thus be computed using a slightly modified version of the VF2 algorithm without the need to switch from graphs to line graphs. Furthermore, multiple alignments of (E-)graphs yield a natural definition of a consensus graph that is ``on average'' present in the input without having to be a subgraph of every (and in fact of any) input graph.

Friday, November 13 2026, 01:00

Tight Bounds for Downward-Closed Prophet and Secretary Problems

Authors: Hanrui Zhang

We study online selection problems subject to arbitrary downward-closed feasibility constraints, under different arrival orders, with different distributional assumptions. We establish the following bounds: (1) $Θ(\log^2 n / \log \log n)$ for the standard (adversarial-order) prophet problem, (2) $Ω(\log n)$ for the (Bayesian, random-order) prophet secretary problem, and by extension, for the standard (prior-free, random-order) secretary problem, and (3) $O(\log n / \log \log n)$ for the free-order prophet problem. Together with complementary bounds established in prior work, these results pin down the asymptotically tight competitive ratios for all these variants of prophet and secretary problems subject to downward-closed constraints. Our results refine and reinforce the big-decisions-first principle by Rubinstein and Singla [2026], establishing a superconstant separation between the random-order and free-order models.

Authors: Hanrui Zhang

We study online selection problems subject to arbitrary downward-closed feasibility constraints, under different arrival orders, with different distributional assumptions. We establish the following bounds: (1) $Θ(\log^2 n / \log \log n)$ for the standard (adversarial-order) prophet problem, (2) $Ω(\log n)$ for the (Bayesian, random-order) prophet secretary problem, and by extension, for the standard (prior-free, random-order) secretary problem, and (3) $O(\log n / \log \log n)$ for the free-order prophet problem. Together with complementary bounds established in prior work, these results pin down the asymptotically tight competitive ratios for all these variants of prophet and secretary problems subject to downward-closed constraints. Our results refine and reinforce the big-decisions-first principle by Rubinstein and Singla [2026], establishing a superconstant separation between the random-order and free-order models.

Friday, November 13 2026, 01:00

Locality Meets Interference: Structure and Scheduling in Radio Multicasting

Authors: Qin Xin

We study multipoint-to-multipoint multicasting in synchronous radio networks with unknown participant locations and unrestricted message aggregation. Let $n$ be the network size, $k$ the number of participants, and $d$ their hop diameter. A supplied distance-two coloring with $C$ colors yields completion within $(C-1)d+1$ slots and at most $nk$ transmissions. In unit disk graphs, coordinates, one-hop neighbor data, and a common degree bound $Δ_0$ suffice for a $15(Δ_0+1)$-color schedule. A known fixed separation bound permits optimal deterministic $Θ(d)$ completion from coordinates alone. Without geometric data, strong selectivity gives $O(d\min\{n,(Δ_0+1)^2\log n\})$ slots when $n$ is known. Independent random transmission gives $O((Δ_0+1)(d+\log(k/η)))$ slots with failure probability at most $η$; for fixed $Δ_0$, this simplifies to $O(d+\log(1/η))$. A star construction shows that the logarithmic term is necessary for this random rule when the degree grows. We extend color schedules to fixed deletions and subgraphs connected over bounded windows of cycles. Separate conditional reductions preserve additive costs under diameter guessing and control parameter-table and cluster preprocessing. Finally, a counterexample shows that pruning and reranking a breadth-first tree can destroy interference-free delivery. For one-hop designated delivery with receiver degree at most $b$, we prove that the optimal schedule length equals the chromatic number of an associated conflict graph and is at most $2b-1$, with matching examples.

Authors: Qin Xin

We study multipoint-to-multipoint multicasting in synchronous radio networks with unknown participant locations and unrestricted message aggregation. Let $n$ be the network size, $k$ the number of participants, and $d$ their hop diameter. A supplied distance-two coloring with $C$ colors yields completion within $(C-1)d+1$ slots and at most $nk$ transmissions. In unit disk graphs, coordinates, one-hop neighbor data, and a common degree bound $Δ_0$ suffice for a $15(Δ_0+1)$-color schedule. A known fixed separation bound permits optimal deterministic $Θ(d)$ completion from coordinates alone. Without geometric data, strong selectivity gives $O(d\min\{n,(Δ_0+1)^2\log n\})$ slots when $n$ is known. Independent random transmission gives $O((Δ_0+1)(d+\log(k/η)))$ slots with failure probability at most $η$; for fixed $Δ_0$, this simplifies to $O(d+\log(1/η))$. A star construction shows that the logarithmic term is necessary for this random rule when the degree grows. We extend color schedules to fixed deletions and subgraphs connected over bounded windows of cycles. Separate conditional reductions preserve additive costs under diameter guessing and control parameter-table and cluster preprocessing. Finally, a counterexample shows that pruning and reranking a breadth-first tree can destroy interference-free delivery. For one-hop designated delivery with receiver degree at most $b$, we prove that the optimal schedule length equals the chromatic number of an associated conflict graph and is at most $2b-1$, with matching examples.

Friday, November 13 2026, 01:00

Explicit Edit Codes via Multicoloring

Authors: Huiling Xie, Zitan Chen

For any fixed integers $q\ge2$ and $t\ge1$, we construct explicit $q$-ary codes of length $n$ that correct any combination of at most $t$ insertions, deletions, and substitutions, with redundancy at most $2t\log_q n+O_{q,t}(\log\log n)$ $q$-ary symbols. This matches the leading redundancy term of Levenshtein's classical existential bound. Building on the standard graph-coloring approach, we use a prime-based multicoloring of the edit-confusion graph to assign multiple colors to each $q$-ary word of length $m

Authors: Huiling Xie, Zitan Chen

For any fixed integers $q\ge2$ and $t\ge1$, we construct explicit $q$-ary codes of length $n$ that correct any combination of at most $t$ insertions, deletions, and substitutions, with redundancy at most $2t\log_q n+O_{q,t}(\log\log n)$ $q$-ary symbols. This matches the leading redundancy term of Levenshtein's classical existential bound. Building on the standard graph-coloring approach, we use a prime-based multicoloring of the edit-confusion graph to assign multiple colors to each $q$-ary word of length $m

Friday, November 13 2026, 01:00

Canonical Truncation for Mixed-Integer and Combinatorial n-Fold Optimization

Authors: Klaus Jansen, Felix Ohnesorge, Björn Schumacher

We consider a special block-structured integer program with $n$ blocks. Every block has a prescribed number of vectors that need to be selected while minimizing the objective $c^{\intercal}x + g(Ax)$, where $c$ is a rational cost vector, $A$ is an integral matrix with $d$ rows, and $g \colon \mathbb{Z}^d \to \mathbb{Q}\cup \{+\infty\}$ is an arbitrary function. We access $g$ via an oracle that minimizes $c^{\intercal}x + g(Ax)$ when all but $O(d)$ variables are fixed. Our algorithm runs in time $(ndΔ)^{O(d)}(Q + \text{poly}(|I|))$ where $Δ$ is the largest absolute value of an entry in $A$, $Q$ is the necessary time to evaluate the oracle for $g$ and $|I|$ is the size of the input. This model is a variation of the model studied by Dadush et al.~[IPCO 23]. Their model has only a single block but individual upper and lower bounds on the variables instead of prescribed block sums. For their model they present an algorithm with running time $t^d(dΔ)^{O(d^2)}Q$ where $t$ is the number of variables. We use our algorithm to solve mixed-integer linear programs with no upper bounds on the variables, a linear objective, and arbitrarily many constraints involving only fractional variables in time $(mΔ)^{O(m)}\text{poly}(|I|)$, where $m$ is the number of constraints involving integral variables and $Δ$ is the largest absolute value of coefficients of integral variables. This improves the dependence on $m$ in the exponent to asymptotically match the dependency in the integer setting. Further, we use our algorithm to generalize the parameterized algorithm for scheduling on uniformly related machines by Rohwedder~[ICALP 25] to obtain an algorithm for multiple knapsack with running time $w_{\max}^{O(d)}\text{poly}(|I|)$ where $w_{\max}$ is the maximum weight of an item and $d$ the number of different weights.

Authors: Klaus Jansen, Felix Ohnesorge, Björn Schumacher

We consider a special block-structured integer program with $n$ blocks. Every block has a prescribed number of vectors that need to be selected while minimizing the objective $c^{\intercal}x + g(Ax)$, where $c$ is a rational cost vector, $A$ is an integral matrix with $d$ rows, and $g \colon \mathbb{Z}^d \to \mathbb{Q}\cup \{+\infty\}$ is an arbitrary function. We access $g$ via an oracle that minimizes $c^{\intercal}x + g(Ax)$ when all but $O(d)$ variables are fixed. Our algorithm runs in time $(ndΔ)^{O(d)}(Q + \text{poly}(|I|))$ where $Δ$ is the largest absolute value of an entry in $A$, $Q$ is the necessary time to evaluate the oracle for $g$ and $|I|$ is the size of the input. This model is a variation of the model studied by Dadush et al.~[IPCO 23]. Their model has only a single block but individual upper and lower bounds on the variables instead of prescribed block sums. For their model they present an algorithm with running time $t^d(dΔ)^{O(d^2)}Q$ where $t$ is the number of variables. We use our algorithm to solve mixed-integer linear programs with no upper bounds on the variables, a linear objective, and arbitrarily many constraints involving only fractional variables in time $(mΔ)^{O(m)}\text{poly}(|I|)$, where $m$ is the number of constraints involving integral variables and $Δ$ is the largest absolute value of coefficients of integral variables. This improves the dependence on $m$ in the exponent to asymptotically match the dependency in the integer setting. Further, we use our algorithm to generalize the parameterized algorithm for scheduling on uniformly related machines by Rohwedder~[ICALP 25] to obtain an algorithm for multiple knapsack with running time $w_{\max}^{O(d)}\text{poly}(|I|)$ where $w_{\max}$ is the maximum weight of an item and $d$ the number of different weights.

Friday, November 13 2026, 01:00

Improved Deterministic Algorithms for $k$-SAT: Generalization of Chains and Recurrence Proof

Authors: Binbin Cui, S. Cliff Liu, Yongrui Yu

We present a deterministic algorithm for 3-SAT running in $1.32736^n$ time. The algorithm is built on a branching-and-chain framework augmented with a polynomial-time expansion procedure that joins short chains through additional clauses to form bounded-size blocks, which impose stronger constraints on the assignment space. After branching, the algorithm chooses between derandomized local search guided by the block structure and enumeration of block assignments together with selected port variables, reducing the remaining formula to 2-SAT. A weighted analysis of branching, partial-assignment, and local-search costs, together with inductive bounds for chains and block interactions, establishes the running-time guarantee. The framework is self-contained and also offers guidance for practical SAT solver design by identifying useful clause-overlap structures and suggesting when to switch between local-search and reduction-based strategies.

Authors: Binbin Cui, S. Cliff Liu, Yongrui Yu

We present a deterministic algorithm for 3-SAT running in $1.32736^n$ time. The algorithm is built on a branching-and-chain framework augmented with a polynomial-time expansion procedure that joins short chains through additional clauses to form bounded-size blocks, which impose stronger constraints on the assignment space. After branching, the algorithm chooses between derandomized local search guided by the block structure and enumeration of block assignments together with selected port variables, reducing the remaining formula to 2-SAT. A weighted analysis of branching, partial-assignment, and local-search costs, together with inductive bounds for chains and block interactions, establishes the running-time guarantee. The framework is self-contained and also offers guidance for practical SAT solver design by identifying useful clause-overlap structures and suggesting when to switch between local-search and reduction-based strategies.

Friday, November 13 2026, 01:00

A Polynomial-Time Sixth-Power Excluded Grid Theorem

Authors: Mohammad Hajiaghayi

We give a polynomial-time construction for a sixth-power excluded-grid theorem. Every graph whose treewidth is at least $g^6$ times a fixed polylogarithmic factor contains a $g\times g$ grid minor that our randomized algorithm finds with constant probability. The algorithm outputs an explicit minor model, and the exponent in its running time is independent of $g$. This improves the ninth-power existence bound of Chuzhoy and Tan~\cite{CT21} and the $g^{98}$ threshold in the polynomial-time construction of Chekuri and Chuzhoy~\cite{CC16}. Quantitative bounds relating treewidth to grid size are central to bidimensionality, developed by Demaine, Hajiaghayi, and collaborators~\cite{DH05,DH08}. The main ingredient is an asymmetric local routing theorem that uses substantially fewer attachment paths than backbone paths. A balanced recursion preserves connectivity by charging deletions to the boundary of a fixed reference cluster. We construct the smaller attachment sets through a routing hierarchy: either it exposes an accessible well-linked boundary, or its quotient graphs contain the desired grid. These arguments implement cluster splitting using approximate cuts and flow computations. An explicit cut-matching construction and a constructive grid-extraction procedure complete the algorithm. We also give a general packing--covering hypothesis for trees meeting three prescribed terminal sets under which the exponent can be reduced below six. A companion Lean development verifies the new construction, its success guarantee, and the composition of its polynomial running time, using explicit interfaces for the published algorithms and standard finite-data operations.

Authors: Mohammad Hajiaghayi

We give a polynomial-time construction for a sixth-power excluded-grid theorem. Every graph whose treewidth is at least $g^6$ times a fixed polylogarithmic factor contains a $g\times g$ grid minor that our randomized algorithm finds with constant probability. The algorithm outputs an explicit minor model, and the exponent in its running time is independent of $g$. This improves the ninth-power existence bound of Chuzhoy and Tan~\cite{CT21} and the $g^{98}$ threshold in the polynomial-time construction of Chekuri and Chuzhoy~\cite{CC16}. Quantitative bounds relating treewidth to grid size are central to bidimensionality, developed by Demaine, Hajiaghayi, and collaborators~\cite{DH05,DH08}. The main ingredient is an asymmetric local routing theorem that uses substantially fewer attachment paths than backbone paths. A balanced recursion preserves connectivity by charging deletions to the boundary of a fixed reference cluster. We construct the smaller attachment sets through a routing hierarchy: either it exposes an accessible well-linked boundary, or its quotient graphs contain the desired grid. These arguments implement cluster splitting using approximate cuts and flow computations. An explicit cut-matching construction and a constructive grid-extraction procedure complete the algorithm. We also give a general packing--covering hypothesis for trees meeting three prescribed terminal sets under which the exponent can be reduced below six. A companion Lean development verifies the new construction, its success guarantee, and the composition of its polynomial running time, using explicit interfaces for the published algorithms and standard finite-data operations.

Friday, November 13 2026, 01:00

Tight Cycle-Cover Inequalities and Improved Approximation for Shortest Common Superstring with Reverse Complements

Authors: Ryosuke Yamano, Tetsuo Shibuya

The Shortest Common Superstring (SCS) problem is a classical idealized model of genome assembly. However, SCS does not account for the double-stranded nature of DNA: sequencing reads may originate from either strand without a known orientation. This motivates a natural variant, Shortest Common Superstring with Reverse Complements (SCS-RC), which asks for a shortest string containing each input string or its reverse complement as a substring. For SCS, the MGREEDY algorithm constructs an optimal cycle cover of the overlap graph and concatenates the strings obtained by breaking each cycle at its minimum-overlap edge. This algorithm was recently shown to achieve a tight $3$-approximation. Its reverse-complement extension, MGREEDY-RC, also constructs an optimal cycle cover of the overlap graph augmented with reverse complements, but its best known approximation guarantee was $3.75$ (CPM 2026). We extend this cycle-cover inequality to SCS-RC and show that MGREEDY-RC achieves a $3$-approximation, which is tight even over the DNA alphabet. The $3$-approximation guarantee for MGREEDY-RC yields a $2.5$-approximation for TGREEDY-RC, which greedily merges the strings derived from the cycle cover. By replacing this greedy merging step with an algorithm based on a reduction to a maximum-weight Hamiltonian path problem constrained to contain a given perfect matching, we obtain a $(22/9+\varepsilon)$-approximation for every fixed $\varepsilon>0$. This improves upon the previous best ratio of $8/3$ (WABI 2026), establishing the best known polynomial-time approximation guarantee for SCS-RC.

Authors: Ryosuke Yamano, Tetsuo Shibuya

The Shortest Common Superstring (SCS) problem is a classical idealized model of genome assembly. However, SCS does not account for the double-stranded nature of DNA: sequencing reads may originate from either strand without a known orientation. This motivates a natural variant, Shortest Common Superstring with Reverse Complements (SCS-RC), which asks for a shortest string containing each input string or its reverse complement as a substring. For SCS, the MGREEDY algorithm constructs an optimal cycle cover of the overlap graph and concatenates the strings obtained by breaking each cycle at its minimum-overlap edge. This algorithm was recently shown to achieve a tight $3$-approximation. Its reverse-complement extension, MGREEDY-RC, also constructs an optimal cycle cover of the overlap graph augmented with reverse complements, but its best known approximation guarantee was $3.75$ (CPM 2026). We extend this cycle-cover inequality to SCS-RC and show that MGREEDY-RC achieves a $3$-approximation, which is tight even over the DNA alphabet. The $3$-approximation guarantee for MGREEDY-RC yields a $2.5$-approximation for TGREEDY-RC, which greedily merges the strings derived from the cycle cover. By replacing this greedy merging step with an algorithm based on a reduction to a maximum-weight Hamiltonian path problem constrained to contain a given perfect matching, we obtain a $(22/9+\varepsilon)$-approximation for every fixed $\varepsilon>0$. This improves upon the previous best ratio of $8/3$ (WABI 2026), establishing the best known polynomial-time approximation guarantee for SCS-RC.

Friday, November 13 2026, 01:00

Hardness, Approximation, and Parameterized Algorithms for Asynchronous Border Minimization

Authors: Giulian Buzatu, Adrian Miclăuş, Alexandru Popa

In the Asynchronous Border Minimization Problem (BMP), a set of DNA probe strings is placed on a grid and synthesized using a common sequence of nucleotide-deposition steps. At each step, every cell may accept or skip the offered nucleotide. The cost is the total number of disagreements between neighboring cells over all steps. We study both free placement, where the probes may be rearranged, and fixed placement, where their positions are given. We show that free-placement BMP on a square grid is NP-complete over three symbols. For one-dimensional BMP, we prove APX-hardness over a binary alphabet, and NP-completeness even when every probe has length two. We also obtain an $8/7$-approximation for the latter restriction. For square grids, we give a randomized $O(n^{1/4}\log n)$-approximation, improving the earlier $O(n^{1/4}\log^2 n)$ guarantee. Both free- and fixed-placement BMP on rectangular grids are fixed-parameter tractable when parameterized only by the allowed border cost.

Authors: Giulian Buzatu, Adrian Miclăuş, Alexandru Popa

In the Asynchronous Border Minimization Problem (BMP), a set of DNA probe strings is placed on a grid and synthesized using a common sequence of nucleotide-deposition steps. At each step, every cell may accept or skip the offered nucleotide. The cost is the total number of disagreements between neighboring cells over all steps. We study both free placement, where the probes may be rearranged, and fixed placement, where their positions are given. We show that free-placement BMP on a square grid is NP-complete over three symbols. For one-dimensional BMP, we prove APX-hardness over a binary alphabet, and NP-completeness even when every probe has length two. We also obtain an $8/7$-approximation for the latter restriction. For square grids, we give a randomized $O(n^{1/4}\log n)$-approximation, improving the earlier $O(n^{1/4}\log^2 n)$ guarantee. Both free- and fixed-placement BMP on rectangular grids are fixed-parameter tractable when parameterized only by the allowed border cost.

Friday, November 13 2026, 01:00

Online Survivable Network Design with Arbitrarily Small Connectivity Loss

Authors: MohammadTaghi Hajiaghayi, Saeed Seddighin

In the online survivable network design problem, a graph with edge costs is known in advance, and requests arrive one at a time. Each request is a pair of vertices $(s,t)$ with a connectivity requirement $k$, and the algorithm must irrevocably buy edges so that $s$ and $t$ are joined by $k$ edge-disjoint paths. The goal is to keep the total cost close to that of an optimal offline solution. Gupta, Krishnaswamy, and Ravi~\cite{GKR12} obtain an $O(k\log^3 n)$-competitive algorithm, where $k$ is the maximum requirement, without relaxing the connectivity requirement. Dehghani, Ehsani, Hajiaghayi, Liaghat, and Seddighin~\cite{DEHLS18} obtain a polylogarithmic competitive ratio when the connectivity requirement is allowed to be relaxed by a factor of $2+\varepsilon$. We show that relaxing the connectivity requirement by a factor of only $1+\varepsilon$ suffices to obtain a polylogarithmic competitive ratio.

Authors: MohammadTaghi Hajiaghayi, Saeed Seddighin

In the online survivable network design problem, a graph with edge costs is known in advance, and requests arrive one at a time. Each request is a pair of vertices $(s,t)$ with a connectivity requirement $k$, and the algorithm must irrevocably buy edges so that $s$ and $t$ are joined by $k$ edge-disjoint paths. The goal is to keep the total cost close to that of an optimal offline solution. Gupta, Krishnaswamy, and Ravi~\cite{GKR12} obtain an $O(k\log^3 n)$-competitive algorithm, where $k$ is the maximum requirement, without relaxing the connectivity requirement. Dehghani, Ehsani, Hajiaghayi, Liaghat, and Seddighin~\cite{DEHLS18} obtain a polylogarithmic competitive ratio when the connectivity requirement is allowed to be relaxed by a factor of $2+\varepsilon$. We show that relaxing the connectivity requirement by a factor of only $1+\varepsilon$ suffices to obtain a polylogarithmic competitive ratio.

Friday, November 13 2026, 01:00

When Elements Fail: Weighted Fault-Tolerant Matroid Bases

Authors: Madhumita Kundu, Soumi Nandi, Saket Saurabh

Given a rank-$r$ matroid on $n$ elements with nonnegative costs and a partition into safe and vulnerable elements, we seek a minimum-cost set that spans after every deletion of at most $k$ selected vulnerable elements. We resolve several open questions posed by Bentert, Fomin, Golovach, and Morelle (ESA 2025) concerning weighted costs, nonuniform failures, and explicitly specified failure scenarios. To this end, we develop two compression methods. For matroids given by an independence oracle, weighted flat-avoidance representatives yield a bounded subset of the ground set containing an optimal solution and a $2^{\mathcal{O}((r+k)^4)}n^{\mathcal{O}(1)}$-time algorithm. This establishes fixed-parameter tractability for weighted costs and nonuniform failures simultaneously. For explicitly represented linear matroids, polynomial certificates allow us to compress families of partial solutions while preserving successful completions under all allowed failures. This yields a deterministic algorithm using $2^{\mathcal{O}(kr(r+k)\log(r+1))}n^{\mathcal{O}(1)}$ field operations over arbitrary fields. Over fixed finite fields, a simpler reduction uses parallel classes. For explicitly listed failure scenarios, we obtain fixed-parameter algorithms parameterized by rank and the number of scenarios. In contrast, the size-budget decision version is W[1]-hard parameterized by rank and maximum scenario size, even on uniform matroids with two-element scenarios. \keywords{Matroids \and Fault tolerance \and Fixed-parameter algorithms \and Representative families}

Authors: Madhumita Kundu, Soumi Nandi, Saket Saurabh

Given a rank-$r$ matroid on $n$ elements with nonnegative costs and a partition into safe and vulnerable elements, we seek a minimum-cost set that spans after every deletion of at most $k$ selected vulnerable elements. We resolve several open questions posed by Bentert, Fomin, Golovach, and Morelle (ESA 2025) concerning weighted costs, nonuniform failures, and explicitly specified failure scenarios. To this end, we develop two compression methods. For matroids given by an independence oracle, weighted flat-avoidance representatives yield a bounded subset of the ground set containing an optimal solution and a $2^{\mathcal{O}((r+k)^4)}n^{\mathcal{O}(1)}$-time algorithm. This establishes fixed-parameter tractability for weighted costs and nonuniform failures simultaneously. For explicitly represented linear matroids, polynomial certificates allow us to compress families of partial solutions while preserving successful completions under all allowed failures. This yields a deterministic algorithm using $2^{\mathcal{O}(kr(r+k)\log(r+1))}n^{\mathcal{O}(1)}$ field operations over arbitrary fields. Over fixed finite fields, a simpler reduction uses parallel classes. For explicitly listed failure scenarios, we obtain fixed-parameter algorithms parameterized by rank and the number of scenarios. In contrast, the size-budget decision version is W[1]-hard parameterized by rank and maximum scenario size, even on uniform matroids with two-element scenarios. \keywords{Matroids \and Fault tolerance \and Fixed-parameter algorithms \and Representative families}

Friday, November 13 2026, 01:00

Optimal-Size Strong Connectivity Preservers under Two Edge Failures in Subcubic Time

Authors: Wanhee Lee, Chenglin Fan

A $k$-edge-failure strong connectivity preserver of a directed graph on $N$ vertices is a spanning subgraph that has the same strongly connected components as the input after each set of at most $k$ edge deletions. The lower-bound construction of Baswana, Choudhary, and Roditty [BCR18], adapted to strong connectivity in [CC20], gives the previous best bound of $Ω(2^kN)$ necessary edges. Hoppenworth, Saranurak, and Wang [HSW25] prove an $O(k4^kN\log N)$ upper bound, leaving a logarithmic gap for fixed $k$. We close this gap by giving an explicit strongly connected graph on $N=2^d$ vertices with $N-1+(N/2)\log_2N$ edges, every one of which is necessary in a two-failure preserver. Together with the upper bound of [HSW25], this proves that the optimal size in the worst case is $Θ_k(N\log N)$ for every fixed $k\geq2$, whereas it is $Θ(N)$ for $k=1$ [GIP20]. The same graph has a one-failure preserver with at most $3(N-1)$ edges. We also give a randomized algorithm that either returns a spanning subgraph or reports failure. For every fixed $c,\varepsilon>0$, with probability at least $1-N^{-c}$ it returns a correct preserver for $k=2$ with $O(N\log N)$ edges, in $\widetilde O(N^{(ω+3)/2+\varepsilon})$ time and $\widetilde O(N^2)$ space. Here $ω$ is the matrix multiplication exponent over a fixed finite field.

Authors: Wanhee Lee, Chenglin Fan

A $k$-edge-failure strong connectivity preserver of a directed graph on $N$ vertices is a spanning subgraph that has the same strongly connected components as the input after each set of at most $k$ edge deletions. The lower-bound construction of Baswana, Choudhary, and Roditty [BCR18], adapted to strong connectivity in [CC20], gives the previous best bound of $Ω(2^kN)$ necessary edges. Hoppenworth, Saranurak, and Wang [HSW25] prove an $O(k4^kN\log N)$ upper bound, leaving a logarithmic gap for fixed $k$. We close this gap by giving an explicit strongly connected graph on $N=2^d$ vertices with $N-1+(N/2)\log_2N$ edges, every one of which is necessary in a two-failure preserver. Together with the upper bound of [HSW25], this proves that the optimal size in the worst case is $Θ_k(N\log N)$ for every fixed $k\geq2$, whereas it is $Θ(N)$ for $k=1$ [GIP20]. The same graph has a one-failure preserver with at most $3(N-1)$ edges. We also give a randomized algorithm that either returns a spanning subgraph or reports failure. For every fixed $c,\varepsilon>0$, with probability at least $1-N^{-c}$ it returns a correct preserver for $k=2$ with $O(N\log N)$ edges, in $\widetilde O(N^{(ω+3)/2+\varepsilon})$ time and $\widetilde O(N^2)$ space. Here $ω$ is the matrix multiplication exponent over a fixed finite field.

Friday, November 13 2026, 01:00

A Binary Search Tree with the Unified Bound

Authors: Yang Cao, Haoran Qi, Hanzhi Wang

We give a deterministic online binary search tree (BST) that satisfies the unified bound of Iacono [SODA 2001], which subsumes the working-set and dynamic-finger bounds: it serves every sequence $X$ of $m$ searches on $n$ keys at total cost $O(m+\mathrm{UB}(X))$, and every single search at cost $O(\log n)$. Previously, no BST was known to satisfy the unified bound, even offline. As a consequence, the dynamic optimality conjecture implies Iacono's conjecture that splay trees satisfy the unified bound. We also show that the unified bound is $O((m+n)(1+\log c_σ))$ on every sequence of searches that avoids a fixed permutation $σ$, where $c_σ$ is the Füredi-Hajnal limit of $σ$. Hence every BST that satisfies the unified bound, including ours, serves such sequences at this cost. For permutations this bound is tight, and it improves the previous upper bound $O(nc_σ^2)$ on the optimal cost, due to Berendsohn, Kozma and Opler [STOC 2024]. Finally, we extend the tree to support insertions and deletions: a sequence $X$ of $m$ operations, starting from $n_0$ keys, costs $O(n_0+m+\mathrm{UB}(X))$ in total, where $\mathrm{UB}$ is now the extension of the unified bound to insertions and deletions by Bădoiu, Cole, Demaine and Iacono [TCS 2007]. Hence, if splay trees are dynamically optimal for searches, insertions and deletions, then they serve every such sequence at this total cost as well.

Authors: Yang Cao, Haoran Qi, Hanzhi Wang

We give a deterministic online binary search tree (BST) that satisfies the unified bound of Iacono [SODA 2001], which subsumes the working-set and dynamic-finger bounds: it serves every sequence $X$ of $m$ searches on $n$ keys at total cost $O(m+\mathrm{UB}(X))$, and every single search at cost $O(\log n)$. Previously, no BST was known to satisfy the unified bound, even offline. As a consequence, the dynamic optimality conjecture implies Iacono's conjecture that splay trees satisfy the unified bound. We also show that the unified bound is $O((m+n)(1+\log c_σ))$ on every sequence of searches that avoids a fixed permutation $σ$, where $c_σ$ is the Füredi-Hajnal limit of $σ$. Hence every BST that satisfies the unified bound, including ours, serves such sequences at this cost. For permutations this bound is tight, and it improves the previous upper bound $O(nc_σ^2)$ on the optimal cost, due to Berendsohn, Kozma and Opler [STOC 2024]. Finally, we extend the tree to support insertions and deletions: a sequence $X$ of $m$ operations, starting from $n_0$ keys, costs $O(n_0+m+\mathrm{UB}(X))$ in total, where $\mathrm{UB}$ is now the extension of the unified bound to insertions and deletions by Bădoiu, Cole, Demaine and Iacono [TCS 2007]. Hence, if splay trees are dynamically optimal for searches, insertions and deletions, then they serve every such sequence at this total cost as well.

Friday, November 13 2026, 01:00

A One-Sided Sublinear-Query Tester for Bounded-Degree 2SAT

Authors: Yuichi Yoshida

We give a one-sided tester for satisfiability of 2CNF formulas in the bounded-degree model. For every degree bound $Δ$, distance parameter $\varepsilon>0$, and $ν\in(0,1/2)$, its query complexity on $n$-variable formulas is $\widetilde O_{\varepsilon,Δ,ν}(n^{1/2+ν})$.

Authors: Yuichi Yoshida

We give a one-sided tester for satisfiability of 2CNF formulas in the bounded-degree model. For every degree bound $Δ$, distance parameter $\varepsilon>0$, and $ν\in(0,1/2)$, its query complexity on $n$-variable formulas is $\widetilde O_{\varepsilon,Δ,ν}(n^{1/2+ν})$.

Friday, November 13 2026, 01:00

Exact and Approximation Algorithms for Computing Smallest SESs and BMSs

Authors: Hiroki Shibata, Hideo Bannai

Compression schemes represent repetitive strings compactly by using repeated substrings. Computing the minimum size of a representation of a given input string is important for compression and for understanding relationships between compression schemes, although this task is NP-hard for several schemes. In this paper, we introduce exact and approximation algorithms for finding smallest substring equation systems (SESs) and bidirectional macro schemes (BMSs). Specifically, for an input string of length $n$, we give deterministic and randomized algorithms for computing a smallest SES using $O^*(4^n)$ and $O^*(2^n)$ time and space, respectively. We also give a deterministic algorithm for computing a smallest BMS using $O^*(2^n)$ time and space. We further give deterministic and randomized $(1+α)$-approximation algorithms for smallest SESs using $O^*(4^{e^{-α}n})$ and $O^*(2^{e^{-α}n})$ time and space, respectively, for every $α\ge 0$. A conversion from SESs to BMSs gives $4(1+α)$-approximation algorithms for smallest BMSs with the same time and space bounds. All time bounds hold in the worst case, and the randomized algorithms return correct answers with high probability. Our algorithms adapt several approaches to exact and approximation algorithms for combinatorial problems, such as dynamic programming over determinant expansions and a greedy algorithm for monotone submodular functions.

Authors: Hiroki Shibata, Hideo Bannai

Compression schemes represent repetitive strings compactly by using repeated substrings. Computing the minimum size of a representation of a given input string is important for compression and for understanding relationships between compression schemes, although this task is NP-hard for several schemes. In this paper, we introduce exact and approximation algorithms for finding smallest substring equation systems (SESs) and bidirectional macro schemes (BMSs). Specifically, for an input string of length $n$, we give deterministic and randomized algorithms for computing a smallest SES using $O^*(4^n)$ and $O^*(2^n)$ time and space, respectively. We also give a deterministic algorithm for computing a smallest BMS using $O^*(2^n)$ time and space. We further give deterministic and randomized $(1+α)$-approximation algorithms for smallest SESs using $O^*(4^{e^{-α}n})$ and $O^*(2^{e^{-α}n})$ time and space, respectively, for every $α\ge 0$. A conversion from SESs to BMSs gives $4(1+α)$-approximation algorithms for smallest BMSs with the same time and space bounds. All time bounds hold in the worst case, and the randomized algorithms return correct answers with high probability. Our algorithms adapt several approaches to exact and approximation algorithms for combinatorial problems, such as dynamic programming over determinant expansions and a greedy algorithm for monotone submodular functions.

Friday, November 13 2026, 01:00

An EPTAS for Max-Cut on Interval Graphs

Authors: Hani Ahmadzadeh, Morteza Alimi

Max-Cut was shown NP-hard on interval graphs by Adhikary, Bose, Mukherjee, and Roy (SoCG 2021). The best previously known approximation ratio for this class, due to Ahn, DeHaan, Kim, and Lee (APPROX 2025), exceeded the Goemans--Williamson ratio of about $0.878$ by only $10^{-34}$. We give a deterministic efficient polynomial-time approximation scheme (EPTAS): for every $\varepsilon\in(0,1)$, a $(1-\varepsilon)$-approximate cut of an unweighted $n$-vertex interval graph can be found in time $2^{O(\varepsilon^{-2}\log^3(2/\varepsilon))}n^2$. The same guarantee and running time hold for Max-Bisection. An FPTAS is impossible unless $\mathrm{P}=\mathrm{NP}$. Our algorithm rests on a structural theorem: every interval graph with $m$ edges can be turned into an interval graph of clique-width $O(\varepsilon^{-2}\log^2(2/\varepsilon))$ by deleting at most $\varepsilon m$ edges. The construction combines endpoint-rank coordinates, length-dependent rounding, and a shifted dyadic hierarchy. Intervals intersecting cell boundaries at their own scales are isolated at small total edge cost; the interactions of remaining long intervals with smaller subproblems can be summarized by counts. The structural theorem also yields a randomized EPTAS via Lampis's scheme for bounded clique-width; our direct dynamic program is deterministic and handles Max-Bisection. Finally, for every $\varepsilon>0$, it is NP-hard to approximate Max-Cut within $84/85+\varepsilon$ on unweighted chordal graphs, which contain interval graphs. Thus an approximation scheme for Max-Cut on chordal graphs would imply $\mathrm{P}=\mathrm{NP}$.

Authors: Hani Ahmadzadeh, Morteza Alimi

Max-Cut was shown NP-hard on interval graphs by Adhikary, Bose, Mukherjee, and Roy (SoCG 2021). The best previously known approximation ratio for this class, due to Ahn, DeHaan, Kim, and Lee (APPROX 2025), exceeded the Goemans--Williamson ratio of about $0.878$ by only $10^{-34}$. We give a deterministic efficient polynomial-time approximation scheme (EPTAS): for every $\varepsilon\in(0,1)$, a $(1-\varepsilon)$-approximate cut of an unweighted $n$-vertex interval graph can be found in time $2^{O(\varepsilon^{-2}\log^3(2/\varepsilon))}n^2$. The same guarantee and running time hold for Max-Bisection. An FPTAS is impossible unless $\mathrm{P}=\mathrm{NP}$. Our algorithm rests on a structural theorem: every interval graph with $m$ edges can be turned into an interval graph of clique-width $O(\varepsilon^{-2}\log^2(2/\varepsilon))$ by deleting at most $\varepsilon m$ edges. The construction combines endpoint-rank coordinates, length-dependent rounding, and a shifted dyadic hierarchy. Intervals intersecting cell boundaries at their own scales are isolated at small total edge cost; the interactions of remaining long intervals with smaller subproblems can be summarized by counts. The structural theorem also yields a randomized EPTAS via Lampis's scheme for bounded clique-width; our direct dynamic program is deterministic and handles Max-Bisection. Finally, for every $\varepsilon>0$, it is NP-hard to approximate Max-Cut within $84/85+\varepsilon$ on unweighted chordal graphs, which contain interval graphs. Thus an approximation scheme for Max-Cut on chordal graphs would imply $\mathrm{P}=\mathrm{NP}$.

Friday, November 13 2026, 01:00

Plurality Consensus for $h$-Majority via First-Repetition Rules

Authors: Niccolò D'Archivio, George Giakkoupis, Frédéric Giroire, Emanuele Natale

In the h-Majority dynamics, each of n agents holds one of k opinions and, in every synchronous round, observes the opinions of h agents chosen independently and uniformly at random and adopts a most frequent one among them, breaking ties uniformly at random. Let M_0 be the initial number of agents holding the most frequent opinion, and Delta_0 its bias, its lead over the second most frequent opinion. We show that, for every sample size h >= 3, if Delta_0 >= L sqrt(M_0 log n) max{1, n/(h^2 M_0)} for a sufficiently large constant L, then h-Majority reaches consensus on the initially most frequent opinion within O((1 + n/(h^2 M_0)) log n) rounds with high probability. This is the time bound proved for the DejaVu dynamics with h samples (arXiv:2604.03648); our bias condition is that of DejaVu when h >= sqrt(n/M_0), and larger by a factor sqrt(n/(h^2 M_0)) otherwise. Our main contribution, the analysis of one round, rests on a comparison principle: among all rules that read the same observations and treat opinions symmetrically, plurality maximizes the difference between the probability of outputting the most frequent opinion and that of outputting any fixed competitor. We apply it to the rule that adopts the first opinion repeated among h-1 observations, and otherwise the h-th observation: its output law is the law of the new opinion of a random agent after one round of DejaVu with h-1 samples. Hence, in expectation, one round of h-Majority widens the gap between the most frequent opinion and each competitor at least as much as that round of DejaVu, whose analysis we use as a black box. Independently and concurrently, d'Amore, Rybicki, Shimizu, Shiraga and Vacus (arXiv:2610.40562) obtained, among other results, the same time bound under the weaker bias condition of DejaVu, with a different analysis of one round and a martingale analysis over many rounds.

Authors: Niccolò D'Archivio, George Giakkoupis, Frédéric Giroire, Emanuele Natale

In the h-Majority dynamics, each of n agents holds one of k opinions and, in every synchronous round, observes the opinions of h agents chosen independently and uniformly at random and adopts a most frequent one among them, breaking ties uniformly at random. Let M_0 be the initial number of agents holding the most frequent opinion, and Delta_0 its bias, its lead over the second most frequent opinion. We show that, for every sample size h >= 3, if Delta_0 >= L sqrt(M_0 log n) max{1, n/(h^2 M_0)} for a sufficiently large constant L, then h-Majority reaches consensus on the initially most frequent opinion within O((1 + n/(h^2 M_0)) log n) rounds with high probability. This is the time bound proved for the DejaVu dynamics with h samples (arXiv:2604.03648); our bias condition is that of DejaVu when h >= sqrt(n/M_0), and larger by a factor sqrt(n/(h^2 M_0)) otherwise. Our main contribution, the analysis of one round, rests on a comparison principle: among all rules that read the same observations and treat opinions symmetrically, plurality maximizes the difference between the probability of outputting the most frequent opinion and that of outputting any fixed competitor. We apply it to the rule that adopts the first opinion repeated among h-1 observations, and otherwise the h-th observation: its output law is the law of the new opinion of a random agent after one round of DejaVu with h-1 samples. Hence, in expectation, one round of h-Majority widens the gap between the most frequent opinion and each competitor at least as much as that round of DejaVu, whose analysis we use as a black box. Independently and concurrently, d'Amore, Rybicki, Shimizu, Shiraga and Vacus (arXiv:2610.40562) obtained, among other results, the same time bound under the weaker bias condition of DejaVu, with a different analysis of one round and a martingale analysis over many rounds.

Friday, November 13 2026, 01:00

Efficient Influence Minimization via Source-Integrated Sampling

Authors: Jingbang Chen, Junyao Wang

Influence minimization by node blocking is a fundamental problem in graph data management that seeks a small set of vertices whose removal minimizes the expected spread from a fixed source set. It supports decisions about where to intervene in large interaction networks to control harmful propagation, such as the spread of rumors and infectious diseases. Under the Independent Cascade (IC) model, selecting blockers requires comparing a large number of candidate sets, and estimating their effects on propagation can be costly. We propose SIG (Source-Integrated Greedy), a sampling framework for efficiently finding high-quality approximate solutions. Our key idea is to group the possible sets of neighbors directly activated by the sources according to their effects on blocker selection. SIG represents these groups in a compact weighted tree and computes their probabilities directly, allowing each sample to account for all source outcomes without enumerating them. We prove that SIG produces unbiased estimates of the lower-bound objective without increasing variance for fixed blocking plans, and establish a high-probability quality guarantee through independent validation. Extensive experiments on real-world networks show that SIG has the lowest mean total running time in all main comparisons, with speedups of up to 583x over the fastest baseline. SIG produces solutions competitive with those of prior methods and achieves the highest influence reduction in multiple settings, including on many of the largest networks, where several baselines exceed the runtime limit. Additional experiments show that SIG maintains consistently low running times as graph size and the blocking budget increase, while delivering stronger blocking solutions at larger budgets.

Authors: Jingbang Chen, Junyao Wang

Influence minimization by node blocking is a fundamental problem in graph data management that seeks a small set of vertices whose removal minimizes the expected spread from a fixed source set. It supports decisions about where to intervene in large interaction networks to control harmful propagation, such as the spread of rumors and infectious diseases. Under the Independent Cascade (IC) model, selecting blockers requires comparing a large number of candidate sets, and estimating their effects on propagation can be costly. We propose SIG (Source-Integrated Greedy), a sampling framework for efficiently finding high-quality approximate solutions. Our key idea is to group the possible sets of neighbors directly activated by the sources according to their effects on blocker selection. SIG represents these groups in a compact weighted tree and computes their probabilities directly, allowing each sample to account for all source outcomes without enumerating them. We prove that SIG produces unbiased estimates of the lower-bound objective without increasing variance for fixed blocking plans, and establish a high-probability quality guarantee through independent validation. Extensive experiments on real-world networks show that SIG has the lowest mean total running time in all main comparisons, with speedups of up to 583x over the fastest baseline. SIG produces solutions competitive with those of prior methods and achieves the highest influence reduction in multiple settings, including on many of the largest networks, where several baselines exceed the runtime limit. Additional experiments show that SIG maintains consistently low running times as graph size and the blocking budget increase, while delivering stronger blocking solutions at larger budgets.

Friday, November 13 2026, 01:00

Postdoc at Bocconi University (apply by January 15, 2027)

from CCI: jobs

A 2-year postdoc position in algorithms at Bocconi. About our theory group: cs.unibocconi.eu/research/theory About life in Milan: lucatrevisan.wordpress.com/category/places/bocconi/ Salary range: EUR45–60k gross per year (researchers moving from abroad can benefit from a very favorable fiscal regime). Website: jobmarket.unibocconi.eu/?id=1013 Email: adam.polak@unibocconi.it

A 2-year postdoc position in algorithms at Bocconi. About our theory group: https://cs.unibocconi.eu/research/theory About life in Milan: https://lucatrevisan.wordpress.com/category/places/bocconi/ Salary range: EUR45–60k gross per year (researchers moving from abroad can benefit from a very favorable fiscal regime).

Website: https://jobmarket.unibocconi.eu/?id=1013
Email: adam.polak@unibocconi.it

By shacharlovett

Thursday, November 12 2026, 23:41

Theory and Algorithms: Tenure-Track Assistant Professor in Computer Science at University of New Mexico (apply by December 5, 2026)

from CCI: jobs

The Department of Computer Science in the School of Engineering at the University of New Mexico invites applications for a full-time, tenure-track Assistant Professor position. We welcome candidates in all areas of computing, particularly those with expertise in quantum computing, theory and algorithms, or cybersecurity. Website: cs.unm.edu/news/2026/10/assistant-professor-position.html Email: saia@unm.edu

The Department of Computer Science in the School of Engineering at the University of New Mexico invites applications for a full-time, tenure-track Assistant Professor position. We welcome candidates in all areas of computing, particularly those with expertise in quantum computing, theory and algorithms, or cybersecurity.

Website: https://cs.unm.edu/news/2026/10/assistant-professor-position.html
Email: saia@unm.edu

By shacharlovett

Thursday, November 12 2026, 18:31

TENURE-TRACK / TENURED FACULTY AT INSAIT at INSAIT – Institute for Computer Science, Artificial Intelligence and Technology (apply by January 31, 2027)

from CCI: jobs

INSAIT invites applications for multiple faculty positions in Theoretical Computer Science, Algorithms, and Complexity. Modelled after top research institutes, INSAIT offers world-class facilities, competitive salaries, generous funding for hiring and travel, and full research autonomy. We welcome applicants of all ranks, including tenured faculty to co-lead and grow the new Theory group. Website: insait.ai/faculty/ Email: […]

INSAIT invites applications for multiple faculty positions in Theoretical Computer Science, Algorithms, and Complexity. Modelled after top research institutes, INSAIT offers world-class facilities, competitive salaries, generous funding for hiring and travel, and full research autonomy. We welcome applicants of all ranks, including tenured faculty to co-lead and grow the new Theory group.

Website: https://insait.ai/faculty/
Email: contact@insait.ai

By shacharlovett

Thursday, November 12 2026, 15:35

POSTDOC AT INSAIT at INSAIT – Institute for Computer Science, Artificial Intelligence and Technology (apply by January 31, 2027)

from CCI: jobs

INSAIT invites applications for multiple postdoctoral positions in Theoretical Computer Science, Algorithms, and Complexity. Modelled after top research institutes, INSAIT offers world-class facilities, competitive salaries, and generous funding for research and travel. We welcome outstanding researchers eager to contribute to the research of the ERC Advanced Grant-funded Theory@INSAIT group. Website: insait.ai/post-doc-at-insait/ Email: contact@insait.ai

INSAIT invites applications for multiple postdoctoral positions in Theoretical Computer Science, Algorithms, and Complexity. Modelled after top research institutes, INSAIT offers world-class facilities, competitive salaries, and generous funding for research and travel. We welcome outstanding researchers eager to contribute to the research of the ERC Advanced Grant-funded Theory@INSAIT group.

Website: https://insait.ai/post-doc-at-insait/
Email: contact@insait.ai

By shacharlovett

Thursday, November 12 2026, 15:33

Statistical-to-computational gaps in central limit theorems for high-dimensional tensors

Authors: Debsurya De, Dmitriy Kunisky

We study high-dimensional central limit theorems (CLTs) as hypothesis testing problems, comparing the number of summands needed for statistical and computational indistinguishability of the normalized sum from the limiting Gaussian distribution as the dimension and number of summands grow together. For the off-diagonal entries of sums of i.i.d. rank-one tensors formed from standard Gaussian vectors (equivalently, the problem of distinguishing Wigner from Wishart tensors), we prove total variation bounds sharpening previous estimates of Mikulincer (2020) and give matching polynomial-time tests. This shows that the statistical and computational thresholds coincide for this CLT, contrary to a proposal by Kunisky, Moore, and Wein (2024). On the other hand, we also show that, for tensors of order at least three, sufficient Gaussian dilution of the summands by adding independent Gaussian noise and rescaling gives a CLT model with a statistical-to-computational gap (in the above setting and ones with more general rank-one summand distributions). To the best of our knowledge, such a phenomenon has not been observed before in any high-dimensional CLT. In particular, in the relevant parameter regimes, we show that a brute-force search achieves strong detection, while polynomials of degree up to $n^{Ω(1)}$ cannot achieve strong separation. The latter lower bounds control low-degree $χ^2$-divergences by a novel method based on cumulant generating functions, and we further illustrate the use of these tools by applying them to masked tensor CLT models where only a prescribed subset of tensor entries is revealed.

Authors: Debsurya De, Dmitriy Kunisky

We study high-dimensional central limit theorems (CLTs) as hypothesis testing problems, comparing the number of summands needed for statistical and computational indistinguishability of the normalized sum from the limiting Gaussian distribution as the dimension and number of summands grow together. For the off-diagonal entries of sums of i.i.d. rank-one tensors formed from standard Gaussian vectors (equivalently, the problem of distinguishing Wigner from Wishart tensors), we prove total variation bounds sharpening previous estimates of Mikulincer (2020) and give matching polynomial-time tests. This shows that the statistical and computational thresholds coincide for this CLT, contrary to a proposal by Kunisky, Moore, and Wein (2024). On the other hand, we also show that, for tensors of order at least three, sufficient Gaussian dilution of the summands by adding independent Gaussian noise and rescaling gives a CLT model with a statistical-to-computational gap (in the above setting and ones with more general rank-one summand distributions). To the best of our knowledge, such a phenomenon has not been observed before in any high-dimensional CLT. In particular, in the relevant parameter regimes, we show that a brute-force search achieves strong detection, while polynomials of degree up to $n^{Ω(1)}$ cannot achieve strong separation. The latter lower bounds control low-degree $χ^2$-divergences by a novel method based on cumulant generating functions, and we further illustrate the use of these tools by applying them to masked tensor CLT models where only a prescribed subset of tensor entries is revealed.

Thursday, November 12 2026, 01:00

Affine Combiners for Reed-Solomon Proximity Gaps

Authors: Alessandro Chiesa, Guy Weissenberg, Eylon Yogev, Kai Zhe Zheng

Reed-Solomon proximity gaps establish soundness guarantees for protocols that test received words through random combinations. For length-$n$ codes over $\mathbb{F}_q$, classical scalar combiners achieve error $O(n/q)$ in the unique-decoding regime, with larger known bounds near the Johnson radius. We prove smaller error bounds using affine combiners, which combine two received words $f,g$ as $f(x)+(α+βx)g(x)$, where $α,β$ are independent uniform elements of $\mathbb{F}_q$. For output degree bound $d=ρn$, fixed $ρ\in(0,1)$ and $0<η<1-\sqrtρ$, and sufficiently large block length $n$, we prove mutual correlated agreement (MCA) with one coordinate of agreement loss at threshold $A=\lceil(\sqrtρ+η)n\rceil$, with error $O_{ρ,η}\left(\frac{1}{q}+\frac{n^2}{q^2}\right)$. The result holds over arbitrary evaluation domains in every finite field, including prime and binary extension fields. The agreement threshold can approach the Johnson threshold, beyond unique decoding. For $q\ge n^2$, our error is $O_{ρ,η}(1/q)$, an asymptotic factor-$n$ separation from scalar combiners: we prove that their worst-case MCA error is $Ω_{ρ,η}(n/q)$ at the same agreement threshold and with the same one-coordinate loss. We also prove that affine combiners incur worst-case error $Ω_{ρ,η}(n/q)$ when no agreement loss is allowed. Thus one coordinate of agreement loss is necessary for our guarantee; this relaxation of standard MCA reduces the guaranteed common agreement fraction by at most $1/n$. We also extend the result to $k$-variate Reed-Muller codes on subfield grids, obtaining MCA error $O_{ρ,η,k}(q^{-k})$ for fixed $ρ,η,k$, sufficiently large $n$, and $q\ge n^{k+1}$, with agreement loss $n^{1-1/k}$.

Authors: Alessandro Chiesa, Guy Weissenberg, Eylon Yogev, Kai Zhe Zheng

Reed-Solomon proximity gaps establish soundness guarantees for protocols that test received words through random combinations. For length-$n$ codes over $\mathbb{F}_q$, classical scalar combiners achieve error $O(n/q)$ in the unique-decoding regime, with larger known bounds near the Johnson radius. We prove smaller error bounds using affine combiners, which combine two received words $f,g$ as $f(x)+(α+βx)g(x)$, where $α,β$ are independent uniform elements of $\mathbb{F}_q$. For output degree bound $d=ρn$, fixed $ρ\in(0,1)$ and $0<η<1-\sqrtρ$, and sufficiently large block length $n$, we prove mutual correlated agreement (MCA) with one coordinate of agreement loss at threshold $A=\lceil(\sqrtρ+η)n\rceil$, with error $O_{ρ,η}\left(\frac{1}{q}+\frac{n^2}{q^2}\right)$. The result holds over arbitrary evaluation domains in every finite field, including prime and binary extension fields. The agreement threshold can approach the Johnson threshold, beyond unique decoding. For $q\ge n^2$, our error is $O_{ρ,η}(1/q)$, an asymptotic factor-$n$ separation from scalar combiners: we prove that their worst-case MCA error is $Ω_{ρ,η}(n/q)$ at the same agreement threshold and with the same one-coordinate loss. We also prove that affine combiners incur worst-case error $Ω_{ρ,η}(n/q)$ when no agreement loss is allowed. Thus one coordinate of agreement loss is necessary for our guarantee; this relaxation of standard MCA reduces the guaranteed common agreement fraction by at most $1/n$. We also extend the result to $k$-variate Reed-Muller codes on subfield grids, obtaining MCA error $O_{ρ,η,k}(q^{-k})$ for fixed $ρ,η,k$, sufficiently large $n$, and $q\ge n^{k+1}$, with agreement loss $n^{1-1/k}$.

Thursday, November 12 2026, 01:00

A Linear-Time Construction on $\mathbb{S}^2$ Achieving a Sublinear Energy Gap on Smale's Seventh Problem

Authors: Rishabh Kabra, Wim Dehaen, Andrea Karlova, Mark Hamilton, Joao Carreira, Maks Ovsjanikov, Jonathan T. Barron

Smale's seventh problem asks for a polynomial-time algorithm that places $N$ points on the unit sphere $\mathbb{S}^2$ with logarithmic energy (the sum of $-\log$ of all pairwise distances) within $O(\log N)$ of the minimum. The minimal energy has an asymptotic expansion with explicit terms of order $N^2$ and $N\log N$ and a linear term whose constant is unknown. Brauchart, Hardin, and Saff conjectured this constant; the resulting expansion, $E_{BHS}(N)$, is known to be an upper bound on the minimum, up to $o(N)$. All previously analyzed constructions on $\mathbb{S}^2$ have energy exceeding $E_{BHS}(N)$ by an amount that grows at least linearly in $N$, i.e. $Ω(N)$, far from the $O(\log N)$ accuracy sought by Smale. We present TerracedZonal, a deterministic construction that generates $N$ points in $Θ(N)$ time and memory. Points are placed on latitude rings whose populations are held constant across bands ("terraces") chosen by a one-dimensional dynamic program, so that adjacent rings within each terrace lock into a staggered triangular lattice. Under a locality assumption, we prove that the energy exceeds $E_{BHS}(N)$ by only $Θ(N^{2/3})$, with an explicit constant; exact evaluation up to $N = 10^9$ confirms this rate. Under the same assumption, its distance from the true minimum is sublinear if the conjecture holds. To our knowledge, this is the first algorithmic construction with a sublinear excess over $E_{BHS}(N)$ of explicit order (proven under the locality assumption), a necessary step toward Smale's seventh problem. With one parameter changed, the same algorithm also gives an equal-weight integration rule on $\mathbb{S}^2$ that runs just as fast and whose error decays at an empirical rate of $N^{-4}$ on smooth functions, versus $N^{-3/2}$ for every previous fast construction we tested.

Authors: Rishabh Kabra, Wim Dehaen, Andrea Karlova, Mark Hamilton, Joao Carreira, Maks Ovsjanikov, Jonathan T. Barron

Smale's seventh problem asks for a polynomial-time algorithm that places $N$ points on the unit sphere $\mathbb{S}^2$ with logarithmic energy (the sum of $-\log$ of all pairwise distances) within $O(\log N)$ of the minimum. The minimal energy has an asymptotic expansion with explicit terms of order $N^2$ and $N\log N$ and a linear term whose constant is unknown. Brauchart, Hardin, and Saff conjectured this constant; the resulting expansion, $E_{BHS}(N)$, is known to be an upper bound on the minimum, up to $o(N)$. All previously analyzed constructions on $\mathbb{S}^2$ have energy exceeding $E_{BHS}(N)$ by an amount that grows at least linearly in $N$, i.e. $Ω(N)$, far from the $O(\log N)$ accuracy sought by Smale. We present TerracedZonal, a deterministic construction that generates $N$ points in $Θ(N)$ time and memory. Points are placed on latitude rings whose populations are held constant across bands ("terraces") chosen by a one-dimensional dynamic program, so that adjacent rings within each terrace lock into a staggered triangular lattice. Under a locality assumption, we prove that the energy exceeds $E_{BHS}(N)$ by only $Θ(N^{2/3})$, with an explicit constant; exact evaluation up to $N = 10^9$ confirms this rate. Under the same assumption, its distance from the true minimum is sublinear if the conjecture holds. To our knowledge, this is the first algorithmic construction with a sublinear excess over $E_{BHS}(N)$ of explicit order (proven under the locality assumption), a necessary step toward Smale's seventh problem. With one parameter changed, the same algorithm also gives an equal-weight integration rule on $\mathbb{S}^2$ that runs just as fast and whose error decays at an empirical rate of $N^{-4}$ on smooth functions, versus $N^{-3/2}$ for every previous fast construction we tested.

Thursday, November 12 2026, 01:00

Delaunay-Based Nearest Neighbor Search with Grid-Accelerated Queries and Parallel Construction

Authors: Pengfei Wang, Shiqing Xin, Changhe Tu

Nearest-neighbor search is a fundamental primitive in computer graphics, with applications in distance field computation, surface reconstruction, and point cloud registration. In many graphics applications, data points lie on a surface manifold embedded in 3D space, whereas queries are distributed throughout the surrounding volume, a setting in which spatial partitioning structures such as k-d trees and grids can become inefficient. Dynamic-programming-based nearest-neighbor search (DP-NNS) offers an alternative: it traverses a successor DAG that records the history of incremental Delaunay construction and often outperforms spatial partitioning structures. However, its sequential construction is slow, and naive parallelization degrades query performance, limiting its practical adoption. We accelerate both queries and construction in DP-NNS. For queries, we introduce a grid-based initialization strategy that precomputes valid starting points on the successor DAG, shortening search paths while preserving exactness and enabling a twofold speedup. For construction, we analyze how parallel insertion affects query cost and develop a parallel algorithm that preserves logarithmic expected query cost under stated geometric assumptions. Our parallel construction algorithm achieves a 10x speedup over the sequential baseline while maintaining nearly the same query cost. We further reorder query points to improve cache locality. Extensive experiments demonstrate that our method outperforms widely used libraries in nearest-neighbor query performance.

Authors: Pengfei Wang, Shiqing Xin, Changhe Tu

Nearest-neighbor search is a fundamental primitive in computer graphics, with applications in distance field computation, surface reconstruction, and point cloud registration. In many graphics applications, data points lie on a surface manifold embedded in 3D space, whereas queries are distributed throughout the surrounding volume, a setting in which spatial partitioning structures such as k-d trees and grids can become inefficient. Dynamic-programming-based nearest-neighbor search (DP-NNS) offers an alternative: it traverses a successor DAG that records the history of incremental Delaunay construction and often outperforms spatial partitioning structures. However, its sequential construction is slow, and naive parallelization degrades query performance, limiting its practical adoption. We accelerate both queries and construction in DP-NNS. For queries, we introduce a grid-based initialization strategy that precomputes valid starting points on the successor DAG, shortening search paths while preserving exactness and enabling a twofold speedup. For construction, we analyze how parallel insertion affects query cost and develop a parallel algorithm that preserves logarithmic expected query cost under stated geometric assumptions. Our parallel construction algorithm achieves a 10x speedup over the sequential baseline while maintaining nearly the same query cost. We further reorder query points to improve cache locality. Extensive experiments demonstrate that our method outperforms widely used libraries in nearest-neighbor query performance.

Thursday, November 12 2026, 01:00

Almost Optimal Lower Bounds under Almost ETH for Approximating Euclidean k-means

Authors: Karthik C. S., Dor Minzer

In the Euclidean $k$-means problem, we are given a set of $n$ points in $\mathbb{R}^d$, and the goal is to find $k$ representative points in space so as to minimize the sum of the squared distances from each point to its closest representative. Algorithmically, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] obtain a $(1+\varepsilon)$-approximation for high-dimensional Euclidean $k$-means in time $2^{(k/\varepsilon)^{O(1)}} \cdot dn^{O(1)}$, which was subsequently improved to $2^{\tilde{O}(k/\varepsilon)} + O(ndk)$ by Feldman, Monemizadeh, and Sohler [SoCG'07]. On the lower-bound side, a recent work of Cohen-Addad, Karthik, Saulpic, and Schwiegelshohn [SoCG'26] proposed the Exponential Time for Expanders Hypothesis (XXH), which asserts exponential-time hardness for $(1+\varepsilon)$-approximating Vertex Cover on vertex expanders, and used it to prove nearly matching running time lower bound whenever $k=\tildeω(1/\varepsilon)$. We observe that XXH does not hold in the parameter regime needed for the above result, and show how to repair their reduction under a more standard assumption. More precisely, we show that assuming the Almost Exponential Time Hypothesis for 3-SAT (Almost ETH), for every constant $β\in(0,1)$, Euclidean $k$-means in $\mathbb{R}^d$ has no $(1+\varepsilon)$-approximation algorithm running in time $2^{(k/\varepsilon)^{1-β}}\text{poly}(n,d)$, whenever $k=\tildeω(1/\varepsilon)$. By modifying the construction, we also obtain a similar result for approximating the Euclidean $k$-median problem.

Authors: Karthik C. S., Dor Minzer

In the Euclidean $k$-means problem, we are given a set of $n$ points in $\mathbb{R}^d$, and the goal is to find $k$ representative points in space so as to minimize the sum of the squared distances from each point to its closest representative. Algorithmically, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] obtain a $(1+\varepsilon)$-approximation for high-dimensional Euclidean $k$-means in time $2^{(k/\varepsilon)^{O(1)}} \cdot dn^{O(1)}$, which was subsequently improved to $2^{\tilde{O}(k/\varepsilon)} + O(ndk)$ by Feldman, Monemizadeh, and Sohler [SoCG'07]. On the lower-bound side, a recent work of Cohen-Addad, Karthik, Saulpic, and Schwiegelshohn [SoCG'26] proposed the Exponential Time for Expanders Hypothesis (XXH), which asserts exponential-time hardness for $(1+\varepsilon)$-approximating Vertex Cover on vertex expanders, and used it to prove nearly matching running time lower bound whenever $k=\tildeω(1/\varepsilon)$. We observe that XXH does not hold in the parameter regime needed for the above result, and show how to repair their reduction under a more standard assumption. More precisely, we show that assuming the Almost Exponential Time Hypothesis for 3-SAT (Almost ETH), for every constant $β\in(0,1)$, Euclidean $k$-means in $\mathbb{R}^d$ has no $(1+\varepsilon)$-approximation algorithm running in time $2^{(k/\varepsilon)^{1-β}}\text{poly}(n,d)$, whenever $k=\tildeω(1/\varepsilon)$. By modifying the construction, we also obtain a similar result for approximating the Euclidean $k$-median problem.

Thursday, November 12 2026, 01:00

Two-Stage Mixed-Integer Quadratic Programming in Polynomial Time

Authors: Alberto Del Pia

Lenstra showed that mixed-integer linear programming can be solved in polynomial time when the number of integer variables is fixed and the number of continuous variables is arbitrary. Mixed-integer quadratic programming (MIQP) can be solved in polynomial time when the total number of variables is fixed. However, since MIQP is NP-hard already without integer variables, the analog of Lenstra's result fails unless P = NP. Hence, polynomial-time algorithms for MIQP with an arbitrary number of continuous variables must exploit additional structure. We study MIQPs with two-stage structure, which underlies Benders decomposition and two-stage stochastic programming. A few linking variables, including all integer variables, are coupled with many blocks of continuous variables that interact only through them, both in the constraints and in the objective. We give a polynomial-time algorithm for MIQP with an arbitrary quadratic objective when the number of linking variables and the block size are fixed, while the number of blocks, and hence the number of variables, is arbitrary. The case with no blocks is MIQP in fixed dimension. None of our assumptions can be dropped. Unless P = NP, the number of linking variables and the block size must both be fixed, the blocks cannot interact or contain integer variables, and large blocks cannot be allowed even if their objective is linear. Unless FPT = W[1], the exponent of the running time must depend on both the number of linking variables and the block size.

Authors: Alberto Del Pia

Lenstra showed that mixed-integer linear programming can be solved in polynomial time when the number of integer variables is fixed and the number of continuous variables is arbitrary. Mixed-integer quadratic programming (MIQP) can be solved in polynomial time when the total number of variables is fixed. However, since MIQP is NP-hard already without integer variables, the analog of Lenstra's result fails unless P = NP. Hence, polynomial-time algorithms for MIQP with an arbitrary number of continuous variables must exploit additional structure. We study MIQPs with two-stage structure, which underlies Benders decomposition and two-stage stochastic programming. A few linking variables, including all integer variables, are coupled with many blocks of continuous variables that interact only through them, both in the constraints and in the objective. We give a polynomial-time algorithm for MIQP with an arbitrary quadratic objective when the number of linking variables and the block size are fixed, while the number of blocks, and hence the number of variables, is arbitrary. The case with no blocks is MIQP in fixed dimension. None of our assumptions can be dropped. Unless P = NP, the number of linking variables and the block size must both be fixed, the blocks cannot interact or contain integer variables, and large blocks cannot be allowed even if their objective is linear. Unless FPT = W[1], the exponent of the running time must depend on both the number of linking variables and the block size.

Thursday, November 12 2026, 01:00

$k$-Product Lists for (Generalized) Feedback Vertex Sets: A Tool for Designing FPT Algorithms

Authors: Rin Saito, Yuma Tamura

Enumerating all inclusion-minimal solutions is a powerful approach to designing fixed-parameter tractable (FPT) algorithms. However, for the \textsc{Feedback Vertex Set} problem, this approach faces a fundamental obstacle, as the number of minimal feedback vertex sets is not, in general, bounded by any function of the solution size~$k$. To overcome this limitation, we introduce a data structure called a \emph{$k$-product list}, which implicitly encodes all minimal solutions while admitting non-minimal ones to maintain efficiency. We present FPT algorithms for computing such lists not only for the \textsc{Feedback Vertex Set} problem but also for its natural generalizations, the \textsc{$\ell$-Forest Deletion Set} problem and the \textsc{$r$-Pseudoforest Deletion Set} problem. Our framework yields simple FPT algorithms for a wide range of variants, providing the first such results in several cases and demonstrating the broad applicability of our approach.

Authors: Rin Saito, Yuma Tamura

Enumerating all inclusion-minimal solutions is a powerful approach to designing fixed-parameter tractable (FPT) algorithms. However, for the \textsc{Feedback Vertex Set} problem, this approach faces a fundamental obstacle, as the number of minimal feedback vertex sets is not, in general, bounded by any function of the solution size~$k$. To overcome this limitation, we introduce a data structure called a \emph{$k$-product list}, which implicitly encodes all minimal solutions while admitting non-minimal ones to maintain efficiency. We present FPT algorithms for computing such lists not only for the \textsc{Feedback Vertex Set} problem but also for its natural generalizations, the \textsc{$\ell$-Forest Deletion Set} problem and the \textsc{$r$-Pseudoforest Deletion Set} problem. Our framework yields simple FPT algorithms for a wide range of variants, providing the first such results in several cases and demonstrating the broad applicability of our approach.

Thursday, November 12 2026, 01:00

Characterizing Local Breadth in Language Generation

Authors: Jon Kleinberg, Fan Wei

Large language models sharpen the validity--breadth trade-off: a generator should avoid hallucinations without collapsing onto a narrow set of valid outputs. We study generation from positive examples of an unknown target in the Kleinberg--Mullainathan model, with arbitrary, possibly adversarial order and no distributional or computational assumptions. Breadth is measured by lower Banach density, which rules out arbitrarily large sparse holes, unlike prefix-based lower asymptotic density. 1. Exact optima in one dimension: For each target $K$ in a countable family, we characterize its optimal lower Banach density $b(K)$ via equivalent combinatorial and topological formulations. One eventually valid generator attains all $b(K)$ simultaneously. Every $α\in[0,1/2]$ occurs as $b(K)$ in some countable family; finite Cantor--Bendixson point rank guarantees $b(K)=1/2$, even without a uniform rank bound. 2. Unavoidable trade-offs in higher dimensions: For $d\ge2$, there need not be a single best generator: optimizing density for one target may worsen the guarantee for another. An explicit family quantifies this intrinsically multi-objective trade-off. 3. Complete characterization of density profiles: We characterize the full region of simultaneously achievable guarantees, with equivalent combinatorial and topological formulations, after excluding geometrically degenerate rectangles. Every prescribed product $\prod_i[0,α_i]$, $α_i\in[0,1/2]$ can occur as an exact achievable region for some language collection. 4. Near-half guarantees and the sharp endpoint: Under a near-logarithmic nondegeneracy condition, one generator achieves filtered density at least $1/2-\varepsilon$ at all finite-point-rank targets, for every $\varepsilon>0$, improving the previous polylogarithmic threshold. The exact $1/2$ endpoint can fail even for a rank-zero singleton family.

Authors: Jon Kleinberg, Fan Wei

Large language models sharpen the validity--breadth trade-off: a generator should avoid hallucinations without collapsing onto a narrow set of valid outputs. We study generation from positive examples of an unknown target in the Kleinberg--Mullainathan model, with arbitrary, possibly adversarial order and no distributional or computational assumptions. Breadth is measured by lower Banach density, which rules out arbitrarily large sparse holes, unlike prefix-based lower asymptotic density. 1. Exact optima in one dimension: For each target $K$ in a countable family, we characterize its optimal lower Banach density $b(K)$ via equivalent combinatorial and topological formulations. One eventually valid generator attains all $b(K)$ simultaneously. Every $α\in[0,1/2]$ occurs as $b(K)$ in some countable family; finite Cantor--Bendixson point rank guarantees $b(K)=1/2$, even without a uniform rank bound. 2. Unavoidable trade-offs in higher dimensions: For $d\ge2$, there need not be a single best generator: optimizing density for one target may worsen the guarantee for another. An explicit family quantifies this intrinsically multi-objective trade-off. 3. Complete characterization of density profiles: We characterize the full region of simultaneously achievable guarantees, with equivalent combinatorial and topological formulations, after excluding geometrically degenerate rectangles. Every prescribed product $\prod_i[0,α_i]$, $α_i\in[0,1/2]$ can occur as an exact achievable region for some language collection. 4. Near-half guarantees and the sharp endpoint: Under a near-logarithmic nondegeneracy condition, one generator achieves filtered density at least $1/2-\varepsilon$ at all finite-point-rank targets, for every $\varepsilon>0$, improving the previous polylogarithmic threshold. The exact $1/2$ endpoint can fail even for a rank-zero singleton family.

Thursday, November 12 2026, 01:00

The Price of Anarchy in Markets for Lemons

Authors: Naveen Durvasula, Tim Roughgarden

We study the worst-case equilibrium gains from trade in simple bilateral trade mechanisms with interdependent valuations. Equilibria in such settings are typically complex, as proposed prices can convey private information held by the proposer. Such equilibria suffer lost gains from trade both due to the inefficiency introduced by screening (as in independent private value settings) and due to adverse selection. We identify broad conditions under which, nevertheless, sequential equilibria are guaranteed to achieve a constant fraction of the first-best gains from trade. Our bounds quantify the central role of rare pessimistic seller signals and rare optimistic buyer signals. The results apply to seller-proposer, buyer-proposer, and random-proposer mechanisms. The key idea in our analysis is to relate equilibrium gains from trade to proposer profits in an auxiliary market with independent private values.

Authors: Naveen Durvasula, Tim Roughgarden

We study the worst-case equilibrium gains from trade in simple bilateral trade mechanisms with interdependent valuations. Equilibria in such settings are typically complex, as proposed prices can convey private information held by the proposer. Such equilibria suffer lost gains from trade both due to the inefficiency introduced by screening (as in independent private value settings) and due to adverse selection. We identify broad conditions under which, nevertheless, sequential equilibria are guaranteed to achieve a constant fraction of the first-best gains from trade. Our bounds quantify the central role of rare pessimistic seller signals and rare optimistic buyer signals. The results apply to seller-proposer, buyer-proposer, and random-proposer mechanisms. The key idea in our analysis is to relate equilibrium gains from trade to proposer profits in an auxiliary market with independent private values.

Thursday, November 12 2026, 01:00

Sublogarithmic Online Graph Exploration

Authors: Julia Baligacs

We study the online graph exploration problem, in which an agent must visit every vertex of an initially unknown weighted graph on $n$ vertices and return to its starting vertex. In this model, a vertex's incident edges and their weights are revealed when it is first visited. Despite extensive research, the best known competitive guarantee for general graphs had remained $\mathcal{O}(\log n)$, a bound already established in 1977 for the greedy TSP heuristic Nearest Neighbor. Subsequent approaches, including hierarchical depth-first search (hDFS) and Blocking, achieve better guarantees on restricted graph classes but have logarithmic or worse competitive ratios on general graphs. This raises the question of whether the logarithmic barrier is inherent and Nearest Neighbor is asymptotically optimal. We answer this question in the negative by giving an $\mathcal{O}(\log n/\log \log n)$-competitive algorithm. Our algorithm is a slight modification of hDFS, and we establish its competitive guarantee with a surprisingly short proof.

Authors: Julia Baligacs

We study the online graph exploration problem, in which an agent must visit every vertex of an initially unknown weighted graph on $n$ vertices and return to its starting vertex. In this model, a vertex's incident edges and their weights are revealed when it is first visited. Despite extensive research, the best known competitive guarantee for general graphs had remained $\mathcal{O}(\log n)$, a bound already established in 1977 for the greedy TSP heuristic Nearest Neighbor. Subsequent approaches, including hierarchical depth-first search (hDFS) and Blocking, achieve better guarantees on restricted graph classes but have logarithmic or worse competitive ratios on general graphs. This raises the question of whether the logarithmic barrier is inherent and Nearest Neighbor is asymptotically optimal. We answer this question in the negative by giving an $\mathcal{O}(\log n/\log \log n)$-competitive algorithm. Our algorithm is a slight modification of hDFS, and we establish its competitive guarantee with a surprisingly short proof.

Thursday, November 12 2026, 01:00

Tripod Reordering: A 33-Queue Layout of Planar Graphs

Authors: Sergey Pupyrev

A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into queues so that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue number. A planar product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood (JACM, 2020) states that every planar graph is a subgraph of the strong product of a graph of simple treewidth at most $3$, a clique $K_3$, and a path. The result has been used to show that the queue number of planar graphs is at most $49$. Recently, Bekos, Gronemann, and Raftopoulou (Algorithmica, 2023) investigated how the general approach based on the product structure can be optimized for planar graphs. They claim that an appropriate reordering of the three vertices associated with each bag in the product structure decomposition, called a tripod, reduces the queue number of planar graphs to $42$. We first highlight a gap in their queue layout algorithm: one of the choices required by the algorithm is not guaranteed to exist. Then we show how to address the limitation and provide an alternative tripod reordering mechanism that reduces the upper bound on the queue number of planar graphs to $33$. As a corollary, bipartite planar graphs have queue number at most $22$.

Authors: Sergey Pupyrev

A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into queues so that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue number. A planar product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood (JACM, 2020) states that every planar graph is a subgraph of the strong product of a graph of simple treewidth at most $3$, a clique $K_3$, and a path. The result has been used to show that the queue number of planar graphs is at most $49$. Recently, Bekos, Gronemann, and Raftopoulou (Algorithmica, 2023) investigated how the general approach based on the product structure can be optimized for planar graphs. They claim that an appropriate reordering of the three vertices associated with each bag in the product structure decomposition, called a tripod, reduces the queue number of planar graphs to $42$. We first highlight a gap in their queue layout algorithm: one of the choices required by the algorithm is not guaranteed to exist. Then we show how to address the limitation and provide an alternative tripod reordering mechanism that reduces the upper bound on the queue number of planar graphs to $33$. As a corollary, bipartite planar graphs have queue number at most $22$.

Thursday, November 12 2026, 01:00

Small Maximum Likelihood on Fixed Cavender-Farris-Neyman Trees: An Approximation Scheme and Limits to Fully Polynomial Approximation

Authors: Mohammad-Hadi Foroughmand-Araabi, Sama Goliaei

Small Maximum Likelihood fits the edge substitution probabilities of a given phylogenetic tree to an observed alignment, summing over the unobserved ancestral states. For the binary Cavender--Farris--Neyman model, we assume a uniform root and an edge-specific substitution probability $p_e\in[0,\tfrac12]$ for each edge $e$, shared across all alignment sites. We give a deterministic polynomial-time approximation scheme. The algorithm achieves a factor of $1+\varepsilon$ for both the negative log likelihood and the loss after subtracting the uniform-root contribution. Its main step fixes ancestral sequences at one periodically spaced class of depths while summing over all remaining ancestors. A posterior entropy bound controls the cost of this restriction, and dynamic programming optimizes the resulting components over a finite parameter grid. We also give a randomized reduction from Clique showing that an FPTAS returning feasible rational edge probabilities would imply $RP=NP$. These results separate fixed-accuracy approximation from fully polynomial approximation for the same fixed-tree model.

Authors: Mohammad-Hadi Foroughmand-Araabi, Sama Goliaei

Small Maximum Likelihood fits the edge substitution probabilities of a given phylogenetic tree to an observed alignment, summing over the unobserved ancestral states. For the binary Cavender--Farris--Neyman model, we assume a uniform root and an edge-specific substitution probability $p_e\in[0,\tfrac12]$ for each edge $e$, shared across all alignment sites. We give a deterministic polynomial-time approximation scheme. The algorithm achieves a factor of $1+\varepsilon$ for both the negative log likelihood and the loss after subtracting the uniform-root contribution. Its main step fixes ancestral sequences at one periodically spaced class of depths while summing over all remaining ancestors. A posterior entropy bound controls the cost of this restriction, and dynamic programming optimizes the resulting components over a finite parameter grid. We also give a randomized reduction from Clique showing that an FPTAS returning feasible rational edge probabilities would imply $RP=NP$. These results separate fixed-accuracy approximation from fully polynomial approximation for the same fixed-tree model.

Thursday, November 12 2026, 01:00

Tokenization: Hardness and Bicriteria Approximation Algorithms

Authors: Atieh Bastan, Xiaoling He, A. Pavan, N. V. Vinodchandran

We study Binary Tokenization: given a corpus of binary strings over $\{0,1\}$ and a vocabulary budget $K$, select at most $K$ substrings (called vocabulary) to minimize the total number of tokens needed to represent the corpus. The primitive symbols $0$ and $1$ are always available, and each document must be segmented separately. Letting $n$ denote the total corpus length and $M$ for the maximum document length, we prove that, for every fixed $\varepsilon\in(0,1)$, Binary Tokenization problem does not have polynomial-time $n^{1-\varepsilon}$-approximation algorithms unless $\mathsf{P}=\mathsf{NP}$. To complement the hardness result, we give a polynomial-time $(γ,γ)$-bicriteria algorithm, with $γ= \sqrt{M}\log n$: The algorithm outputs a vocabulary of size at most $γK$ with number of tokens at most $γ{\rm OPT}_K$, where ${\rm OPT}_K$ is the optimal number of tokens under the original budget $K$. We also study a storage-budget variant, in which the sum of the selected substrings' lengths is bounded by $B$. For $2\le B

Authors: Atieh Bastan, Xiaoling He, A. Pavan, N. V. Vinodchandran

We study Binary Tokenization: given a corpus of binary strings over $\{0,1\}$ and a vocabulary budget $K$, select at most $K$ substrings (called vocabulary) to minimize the total number of tokens needed to represent the corpus. The primitive symbols $0$ and $1$ are always available, and each document must be segmented separately. Letting $n$ denote the total corpus length and $M$ for the maximum document length, we prove that, for every fixed $\varepsilon\in(0,1)$, Binary Tokenization problem does not have polynomial-time $n^{1-\varepsilon}$-approximation algorithms unless $\mathsf{P}=\mathsf{NP}$. To complement the hardness result, we give a polynomial-time $(γ,γ)$-bicriteria algorithm, with $γ= \sqrt{M}\log n$: The algorithm outputs a vocabulary of size at most $γK$ with number of tokens at most $γ{\rm OPT}_K$, where ${\rm OPT}_K$ is the optimal number of tokens under the original budget $K$. We also study a storage-budget variant, in which the sum of the selected substrings' lengths is bounded by $B$. For $2\le B

Thursday, November 12 2026, 01:00

Bounded Knapsack in Truly Subquadratic Time

Authors: Huanyi Zhou

Bounded Knapsack asks how many copies of each item to take to maximize profit under a capacity constraint. Given $n$ records with binary-encoded multiplicities, we return optimal counts in randomized time with a fixed power saving in the maximum item weight $w_{\max}$ or maximum item profit $p_{\max}$. Fix $1<ρ<2$ and suppose integer $(\min,+)$-convolution on length-$m$ arrays can be computed with failure probability at most $η$ in $\widetilde O(m^ρ)$ time, up to polynomial factors in the entry bit length and $\log(1/η)$. Put $f(ρ)=4(3-ρ)/(8-3ρ).$ For any fixed $ε>0$, on inputs whose integers have magnitude polynomial in $n+\min(w_{\max},p_{\max})$, our algorithm runs in time $$\widetilde O\bigl(n\min(w_{\max},p_{\max})^{f(ρ)+ε}\bigr).$$ It returns optimal counts with high probability. If convolution saves $δ=2-ρ$ in its exponent, knapsack retains $δ/(1+3δ/2)$. With the integer convolution exponent $1.9601397040925$, a separate parameter choice gives $\widetilde O(n+\min(w_{\max},p_{\max})^{1.961})$, including for 0-1 Knapsack. We keep the copies compressed. Witnessed convolutions generate residual queries, and local linear programs filter them by cost. Resilient lattice cores allow recursive states to coalesce. Progression reserves support rounding, and a thin affordable fiber yields a lift that reduces the normalized column volume. This volume bounds the number of LP regions a grid lookup examines.

Authors: Huanyi Zhou

Bounded Knapsack asks how many copies of each item to take to maximize profit under a capacity constraint. Given $n$ records with binary-encoded multiplicities, we return optimal counts in randomized time with a fixed power saving in the maximum item weight $w_{\max}$ or maximum item profit $p_{\max}$. Fix $1<ρ<2$ and suppose integer $(\min,+)$-convolution on length-$m$ arrays can be computed with failure probability at most $η$ in $\widetilde O(m^ρ)$ time, up to polynomial factors in the entry bit length and $\log(1/η)$. Put $f(ρ)=4(3-ρ)/(8-3ρ).$ For any fixed $ε>0$, on inputs whose integers have magnitude polynomial in $n+\min(w_{\max},p_{\max})$, our algorithm runs in time $$\widetilde O\bigl(n\min(w_{\max},p_{\max})^{f(ρ)+ε}\bigr).$$ It returns optimal counts with high probability. If convolution saves $δ=2-ρ$ in its exponent, knapsack retains $δ/(1+3δ/2)$. With the integer convolution exponent $1.9601397040925$, a separate parameter choice gives $\widetilde O(n+\min(w_{\max},p_{\max})^{1.961})$, including for 0-1 Knapsack. We keep the copies compressed. Witnessed convolutions generate residual queries, and local linear programs filter them by cost. Resilient lattice cores allow recursive states to coalesce. Progression reserves support rounding, and a thin affordable fiber yields a lift that reduces the normalized column volume. This volume bounds the number of LP regions a grid lookup examines.

Thursday, November 12 2026, 01:00

Sharp Lower Bounds for Parallel Log-Concave Sampling

Authors: Evan Dogariu

How many rounds of parallel queries are needed to sample from a smooth, strongly log-concave distribution on $\mathbb{R}^d$? For every fixed condition number $κ\in (1, \infty)$, we prove that any sampler making polynomially many queries per round requires $Ω_κ(\log d)$ rounds to achieve constant total variation error in the worst case. The lower bound holds even if queries return derivatives of all orders. Together with the first-order parallel Picard algorithm of \cite{zhou2025parallel}, this establishes a depth complexity of $Θ_κ(\log d)$ for parallel log-concave sampling with $\operatorname{poly}(d)$ width. We then study centered Gaussian targets to understand how much width is needed at a given depth. From \cite{chewi2023query,zhou2026complexitygaussiansampling}, it is known that either one round of width $d$ or $\mathcal{O}(\sqrtκ \log d)$ rounds of constant width suffice for first-order Gaussian sampling. At fixed $κ$ we characterize the depth-width tradeoff interpolating between these regimes: when using $ \lceil c \log d\rceil$ rounds, if $0 < c < c_\star(κ)$ then the optimal width at constant TV error is $\widetildeΘ(d^{1-c/c_\star(κ)})$, while if $c \geq c_\star(κ)$ then it is $Θ_κ(1)$. Here, $c_\star(κ)$ is an explicit constant with $c_\star(κ) \sim \sqrtκ / 8$ as $κ\to \infty$.

Authors: Evan Dogariu

How many rounds of parallel queries are needed to sample from a smooth, strongly log-concave distribution on $\mathbb{R}^d$? For every fixed condition number $κ\in (1, \infty)$, we prove that any sampler making polynomially many queries per round requires $Ω_κ(\log d)$ rounds to achieve constant total variation error in the worst case. The lower bound holds even if queries return derivatives of all orders. Together with the first-order parallel Picard algorithm of \cite{zhou2025parallel}, this establishes a depth complexity of $Θ_κ(\log d)$ for parallel log-concave sampling with $\operatorname{poly}(d)$ width. We then study centered Gaussian targets to understand how much width is needed at a given depth. From \cite{chewi2023query,zhou2026complexitygaussiansampling}, it is known that either one round of width $d$ or $\mathcal{O}(\sqrtκ \log d)$ rounds of constant width suffice for first-order Gaussian sampling. At fixed $κ$ we characterize the depth-width tradeoff interpolating between these regimes: when using $ \lceil c \log d\rceil$ rounds, if $0 < c < c_\star(κ)$ then the optimal width at constant TV error is $\widetildeΘ(d^{1-c/c_\star(κ)})$, while if $c \geq c_\star(κ)$ then it is $Θ_κ(1)$. Here, $c_\star(κ)$ is an explicit constant with $c_\star(κ) \sim \sqrtκ / 8$ as $κ\to \infty$.

Thursday, November 12 2026, 01:00

Conflict-Free Cuts: Exploring the Boundaries of Tractability

Authors: Subrahmanyam Kalyanasundaram, Subodh Kumar

A conflict-free cut of a host graph $G$ is an edge cut containing no conflicting pair of edges. The conflicts are specified by a conflict graph $\widehat G$ whose vertex set is $E(G)$. The conflict-free cut problem asks if the given instance $(G, \widehat G)$ has a conflict-free cut. When restricted to planar $G$ and 1-regular $\widehat G$, the problem is polynomial time solvable when $G$ has maximum degree 4, but NP-complete already at maximum degree 5. We study the complexity of conflict-free cut problem under restrictions on the degrees of $G$, the components of $\widehat G$, and the locality of the conflicts. A conflict is local if its two edges share an endpoint in $G$. We prove the following results. * For $Δ(\widehat G)\le1$, a directed-cycle formulation gives a polynomial-time algorithm when $|E(G)|\le2|V(G)|$, hence for maximum degree four. The problem is NP-complete on $5$-regular hosts with exactly $1$-regular conflicts. * For $Δ(\widehat G)\le1$, we give a constructive FPT algorithm in $s_4(G)=\sum_v\max\{0,d_G(v)-4\}$, single-exponential for planar hosts. For arbitrary conflicts, we give an XP algorithm in $η=\max\{0,|E(\widehat G)|-|V(G)|\}$. * On $4$-regular hosts, local exactly $2$-regular conflicts with components $C_3,C_4$ give NP-completeness. For local conflicts at maximum degree four, pair or triangle components separately permit polynomial algorithms, but mixing them gives NP-completeness. We classify cases where $\widehat G$ is a union of cliques. * An $a$-COC is a vertex set whose deletion leaves components of order at most $a$. For local conflicts, the problem is FPT in the minimum $2$-COC size, but NP-complete with a supplied $3$-COC of size six. Without locality, hardness holds with a supplied $2$-COC of size six. Both hardness results use exactly $1$-regular conflicts.

Authors: Subrahmanyam Kalyanasundaram, Subodh Kumar

A conflict-free cut of a host graph $G$ is an edge cut containing no conflicting pair of edges. The conflicts are specified by a conflict graph $\widehat G$ whose vertex set is $E(G)$. The conflict-free cut problem asks if the given instance $(G, \widehat G)$ has a conflict-free cut. When restricted to planar $G$ and 1-regular $\widehat G$, the problem is polynomial time solvable when $G$ has maximum degree 4, but NP-complete already at maximum degree 5. We study the complexity of conflict-free cut problem under restrictions on the degrees of $G$, the components of $\widehat G$, and the locality of the conflicts. A conflict is local if its two edges share an endpoint in $G$. We prove the following results. * For $Δ(\widehat G)\le1$, a directed-cycle formulation gives a polynomial-time algorithm when $|E(G)|\le2|V(G)|$, hence for maximum degree four. The problem is NP-complete on $5$-regular hosts with exactly $1$-regular conflicts. * For $Δ(\widehat G)\le1$, we give a constructive FPT algorithm in $s_4(G)=\sum_v\max\{0,d_G(v)-4\}$, single-exponential for planar hosts. For arbitrary conflicts, we give an XP algorithm in $η=\max\{0,|E(\widehat G)|-|V(G)|\}$. * On $4$-regular hosts, local exactly $2$-regular conflicts with components $C_3,C_4$ give NP-completeness. For local conflicts at maximum degree four, pair or triangle components separately permit polynomial algorithms, but mixing them gives NP-completeness. We classify cases where $\widehat G$ is a union of cliques. * An $a$-COC is a vertex set whose deletion leaves components of order at most $a$. For local conflicts, the problem is FPT in the minimum $2$-COC size, but NP-complete with a supplied $3$-COC of size six. Without locality, hardness holds with a supplied $2$-COC of size six. Both hardness results use exactly $1$-regular conflicts.

Thursday, November 12 2026, 01:00

A Faster Scaling Algorithm for Maximum Weight Matroid Intersection

Authors: Alina Ene, Huy L. Nguyen

We consider the maximum weight matroid intersection problem. In this problem, we are given two matroids $M_{1}=(E,\mathcal{I}_{1})$ and $M_{2}=(E,\mathcal{I}_{2})$ and integer weights $w(e)$ on the elements $e\in E$, and the goal is to find a maximum-weight common independent set. We give a randomized algorithm that constructs an optimal solution using $\widetilde{O}(nr\log W)$ queries to the independence oracles of the matroids, where $n$ is the size of the ground set $E$, $r$ is the minimum rank of the two matroids, and $W$ is the maximum weight.

Authors: Alina Ene, Huy L. Nguyen

We consider the maximum weight matroid intersection problem. In this problem, we are given two matroids $M_{1}=(E,\mathcal{I}_{1})$ and $M_{2}=(E,\mathcal{I}_{2})$ and integer weights $w(e)$ on the elements $e\in E$, and the goal is to find a maximum-weight common independent set. We give a randomized algorithm that constructs an optimal solution using $\widetilde{O}(nr\log W)$ queries to the independence oracles of the matroids, where $n$ is the size of the ground set $E$, $r$ is the minimum rank of the two matroids, and $W$ is the maximum weight.

Thursday, November 12 2026, 01:00

Tree Decomposition of Product and Nest Graphs for Assortment Optimization under the Generalized Nested Logit Model

Authors: Yuheng Wang, Paat Rusmevichientong, Huseyin Topaloglu

The generalized nested logit model extends the nested logit model by allowing each product to belong to multiple nests. The overlapping nests in the generalized nested logit model can capture complex substitution patterns, and the model has found applications in many areas. We study the assortment optimization problem under the generalized nested logit model and prove that it is NP-hard when there are only two nests or when any two distinct nests have at most one product in common. Motivated by these complexity results, we introduce novel concepts of product and nest graphs that capture which products are included in which nests. We use the tree decompositions of the graphs to characterize when the assortment optimization problem admits efficient solution methods. When the treewidth of a product graph is bounded, the assortment optimization problem can be solved in polynomial time. If the treewidth of the product graph is unbounded but the treewidth of the nest graph is bounded, then there exists a fully polynomial-time approximation scheme. These results represent the first polynomial-time algorithms for assortment optimization under the generalized nested logit model. Our numerical experiments on synthetic and real datasets show that the generalized nested logit model provides a better fit to the data than many existing models in the literature.

Authors: Yuheng Wang, Paat Rusmevichientong, Huseyin Topaloglu

The generalized nested logit model extends the nested logit model by allowing each product to belong to multiple nests. The overlapping nests in the generalized nested logit model can capture complex substitution patterns, and the model has found applications in many areas. We study the assortment optimization problem under the generalized nested logit model and prove that it is NP-hard when there are only two nests or when any two distinct nests have at most one product in common. Motivated by these complexity results, we introduce novel concepts of product and nest graphs that capture which products are included in which nests. We use the tree decompositions of the graphs to characterize when the assortment optimization problem admits efficient solution methods. When the treewidth of a product graph is bounded, the assortment optimization problem can be solved in polynomial time. If the treewidth of the product graph is unbounded but the treewidth of the nest graph is bounded, then there exists a fully polynomial-time approximation scheme. These results represent the first polynomial-time algorithms for assortment optimization under the generalized nested logit model. Our numerical experiments on synthetic and real datasets show that the generalized nested logit model provides a better fit to the data than many existing models in the literature.

Thursday, November 12 2026, 01:00

Multiple-Choice Knapsack with Small Items

Authors: Jakub Pawlewicz

The Multiple-Choice Knapsack problem is a generalization of the Knapsack problem in which items are partitioned into disjoint classes and exactly one item must be selected from each class. Besides the standard dynamic program running in time $\mathcal{O}(nc)$, where $n$ is the total number of items and $c$ is the knapsack capacity, no other pseudopolynomial-time algorithm is currently known. We present an $\mathcal{O}(n+w_{\max}^4)$-time algorithm, where $w_{\max}=\max_i (w_{in_i}-w_{i1})$ is the maximum weight range within a class.

Authors: Jakub Pawlewicz

The Multiple-Choice Knapsack problem is a generalization of the Knapsack problem in which items are partitioned into disjoint classes and exactly one item must be selected from each class. Besides the standard dynamic program running in time $\mathcal{O}(nc)$, where $n$ is the total number of items and $c$ is the knapsack capacity, no other pseudopolynomial-time algorithm is currently known. We present an $\mathcal{O}(n+w_{\max}^4)$-time algorithm, where $w_{\max}=\max_i (w_{in_i}-w_{i1})$ is the maximum weight range within a class.

Thursday, November 12 2026, 01:00

Minimizing Monotone Norms in Fractional Load Balancing

Authors: Daniel Blankenburg, Antonia Ellerbrock, Thomas Kesselheim, Jens Vygen

We devise a fully polynomial time approximation scheme for minimizing a monotone norm $\left\lVert\cdot\right\rVert$ over a convex set $X\subseteq \mathbb{R}^d_{\geq 0}$. We study the multi-customer setting, where $X=\sum_{i=1}^n X_i$ and the solution set $X_i\subseteq \mathbb{R}^d_{\geq 0}$ is given by a linear minimization oracle for each customer $i$. Our algorithm computes, for any given $\varepsilon>0$, a $(1+\varepsilon)$-approximate solution. With high probability, it uses $\tilde{\mathcal{O}}\left( \varepsilon^{-2}(n+d) \right)$ single-customer linear minimization oracle calls and the same number of generalized KL projections to the dual space $Y=\{y\in \mathbb{R}^d_{\geq 0}: \left\lVert y \right\rVert^* \le 1\}$. We assume no prior knowledge on $X$, and our running time depends only on $n$, $d$, and $\varepsilon$. Our result is robust in the sense that it allows for approximate linear minimization oracles and approximate KL projections. We show that the (approximate) KL projection can be performed efficiently for large classes of monotone norms, demonstrating the applicability of our approach. While such a result (with exact projection) is well-known for the $\ell_{\infty}$ norm via multiplicative weight updates and was recently obtained for ordered norms via a smooth norm approximation and a novel randomization, existing techniques do not extend to general monotone norms. In particular, a key feature used in previous proofs was dual price stability, which was derived from a global contraction property of the KL projection to the dual space $Y$. We show that this property does not extend to general monotone norms. Therefore, we use a different price mechanism, based on the well-known mirror descent paradigm, for which a local contraction property holds and guarantees stability.

Authors: Daniel Blankenburg, Antonia Ellerbrock, Thomas Kesselheim, Jens Vygen

We devise a fully polynomial time approximation scheme for minimizing a monotone norm $\left\lVert\cdot\right\rVert$ over a convex set $X\subseteq \mathbb{R}^d_{\geq 0}$. We study the multi-customer setting, where $X=\sum_{i=1}^n X_i$ and the solution set $X_i\subseteq \mathbb{R}^d_{\geq 0}$ is given by a linear minimization oracle for each customer $i$. Our algorithm computes, for any given $\varepsilon>0$, a $(1+\varepsilon)$-approximate solution. With high probability, it uses $\tilde{\mathcal{O}}\left( \varepsilon^{-2}(n+d) \right)$ single-customer linear minimization oracle calls and the same number of generalized KL projections to the dual space $Y=\{y\in \mathbb{R}^d_{\geq 0}: \left\lVert y \right\rVert^* \le 1\}$. We assume no prior knowledge on $X$, and our running time depends only on $n$, $d$, and $\varepsilon$. Our result is robust in the sense that it allows for approximate linear minimization oracles and approximate KL projections. We show that the (approximate) KL projection can be performed efficiently for large classes of monotone norms, demonstrating the applicability of our approach. While such a result (with exact projection) is well-known for the $\ell_{\infty}$ norm via multiplicative weight updates and was recently obtained for ordered norms via a smooth norm approximation and a novel randomization, existing techniques do not extend to general monotone norms. In particular, a key feature used in previous proofs was dual price stability, which was derived from a global contraction property of the KL projection to the dual space $Y$. We show that this property does not extend to general monotone norms. Therefore, we use a different price mechanism, based on the well-known mirror descent paradigm, for which a local contraction property holds and guarantees stability.

Thursday, November 12 2026, 01:00

Beyond Worst-Case Online Algorithms via Input Subsampling

Authors: Siddhartha Banerjee, Tianle Jiang, Kamesh Munagala

We study whether classical competitive ratios survive input subsampling, when both the online algorithm and the offline optimum are evaluated on the same subsampled input. We consider the canonical problems of prophet inequalities and ski-rental. In these settings, for any single subsampling rate, the classical worst-case ratios remain unchanged. We ask a stronger question: can one adversarial input remain worst-case at two distinct rates $p>q$? For prophet inequalities, in the model where the policy knows the sampling rate and learns item survival online, every instance has competitive ratio strictly below $2$ at one of the two rates, and we determine the tight simultaneous guarantee. For ski rental, in the fluid model where sampling scales the horizon, the answer depends on $q/p$. Against adversarial distributions over horizons, one unrestricted randomized algorithm achieves a strict simultaneous improvement over $e/(e-1)$ for every distribution exactly when $q/p$ is below approximately $0.34$. Requiring the algorithm to buy by the breakeven time lowers this threshold to approximately $0.31$, showing that unlike classical ski-rental algorithms, buying later can provably help in this setting. Our results therefore reveal a new form of beyond-worst-case fragility under input subsampling: instances that are worst-case at one rate need not remain hard at two rates simultaneously.

Authors: Siddhartha Banerjee, Tianle Jiang, Kamesh Munagala

We study whether classical competitive ratios survive input subsampling, when both the online algorithm and the offline optimum are evaluated on the same subsampled input. We consider the canonical problems of prophet inequalities and ski-rental. In these settings, for any single subsampling rate, the classical worst-case ratios remain unchanged. We ask a stronger question: can one adversarial input remain worst-case at two distinct rates $p>q$? For prophet inequalities, in the model where the policy knows the sampling rate and learns item survival online, every instance has competitive ratio strictly below $2$ at one of the two rates, and we determine the tight simultaneous guarantee. For ski rental, in the fluid model where sampling scales the horizon, the answer depends on $q/p$. Against adversarial distributions over horizons, one unrestricted randomized algorithm achieves a strict simultaneous improvement over $e/(e-1)$ for every distribution exactly when $q/p$ is below approximately $0.34$. Requiring the algorithm to buy by the breakeven time lowers this threshold to approximately $0.31$, showing that unlike classical ski-rental algorithms, buying later can provably help in this setting. Our results therefore reveal a new form of beyond-worst-case fragility under input subsampling: instances that are worst-case at one rate need not remain hard at two rates simultaneously.

Thursday, November 12 2026, 01:00

Optimal-Time Independent Subset Sampling over Acyclic Joins

Authors: Xuyang Liu, Zijian Zhang, Zhen Li

Join results can greatly exceed their inputs even when applications need only a small random subset. We study sampling that includes each result independently with a prescribed probability given by the product of its input-tuple weights (PRODUCT). For fixed full acyclic joins with static weights, we achieve input-linear preprocessing and index space together with output-linear expected sampling work, eliminating logarithmic factors from previous bounds. A shared join-tree index supports weighted draws over the entire join and locates high-probability results. Direct Bernoulli trials handle high-probability results, allowing Poisson thinning to generate the remaining independent inclusion decisions with bounded overhead. Our exact sampler uses expected $O(N)$ preprocessing, $O(N)$ index space, and expected $O(1+mμ)$ time per call including output construction. Here $N$ is the input size, $m$ counts relation occurrences, and $μ$ is the expected sample size. The guarantees hold on a real random-access machine with unit-cost exact arithmetic and logarithms, continuous uniform randomness, and expected-time hashing. The time bounds are optimal, matching input reading and explicit output writing. Our floating-point prototype has lower sampling latency than the fastest applicable baseline in 17 of 18 PRODUCT settings on three queries over the STATS benchmark. Construction savings on large joins can offset indexed materialization's faster sampling over a limited number of calls. On a real-data path at $μ=100$, our sampler achieves a $20.5\times$ speedup over indexed materialization for construction plus 300 calls, measured after common data preparation and loading.

Authors: Xuyang Liu, Zijian Zhang, Zhen Li

Join results can greatly exceed their inputs even when applications need only a small random subset. We study sampling that includes each result independently with a prescribed probability given by the product of its input-tuple weights (PRODUCT). For fixed full acyclic joins with static weights, we achieve input-linear preprocessing and index space together with output-linear expected sampling work, eliminating logarithmic factors from previous bounds. A shared join-tree index supports weighted draws over the entire join and locates high-probability results. Direct Bernoulli trials handle high-probability results, allowing Poisson thinning to generate the remaining independent inclusion decisions with bounded overhead. Our exact sampler uses expected $O(N)$ preprocessing, $O(N)$ index space, and expected $O(1+mμ)$ time per call including output construction. Here $N$ is the input size, $m$ counts relation occurrences, and $μ$ is the expected sample size. The guarantees hold on a real random-access machine with unit-cost exact arithmetic and logarithms, continuous uniform randomness, and expected-time hashing. The time bounds are optimal, matching input reading and explicit output writing. Our floating-point prototype has lower sampling latency than the fastest applicable baseline in 17 of 18 PRODUCT settings on three queries over the STATS benchmark. Construction savings on large joins can offset indexed materialization's faster sampling over a limited number of calls. On a real-data path at $μ=100$, our sampler achieves a $20.5\times$ speedup over indexed materialization for construction plus 300 calls, measured after common data preparation and loading.

Thursday, November 12 2026, 01:00

Online Vertex Connectivity Augmentation

Authors: Yaowei Long, Sepideh Mahabadi, Jakub Tarnawski

Network design problems form a classic subfield of online algorithms, but almost nothing is known about their more challenging vertex-connectivity variants. We initiate the study of online vertex-connectivity network design for arbitrary demand pairs by considering the fundamental problem of connectivity augmentation. In an instance of \emph{Online Vertex Connectivity Augmentation} (OVCA), we are given a $k$-vertex-connected graph $F$ and a set of additional purchasable edges (with costs). Demand pairs $(s_1,t_1),(s_2,t_2),\dots$ then arrive online. Upon each arrival, the algorithm must irrevocably purchase edges so that $s_i$ and $t_i$ become $(k{+}1)$-vertex-connected in the union of $F$ with all edges purchased so far. Our main contribution is an $O((k+\log n)\log n)$-competitive online algorithm for OVCA. We also give an offline $O(k+\log n)$-approximation algorithm. We obtain our algorithms via a structural dichotomy on the minimum vertex cuts of $F$. We note that some dependence on $k$ in the approximation ratio is necessary -- i.e., the problem is hard to approximate within any $\mathrm{polylog}(n)$ factor independent of $k$ -- which shows a separation from the edge-connectivity setting. Finally, specializing to $k=1$, our methods yield the first polylogarithmic-competitive online algorithm for a vertex-connectivity survivable network design problem, namely an $\widetilde O(\log^3 n)$-competitive algorithm for online $\{1,2\}$-VC-SNDP (i.e., an SNDP variant where each demand pair comes with a vertex-connectivity requirement of $1$ or $2$).

Authors: Yaowei Long, Sepideh Mahabadi, Jakub Tarnawski

Network design problems form a classic subfield of online algorithms, but almost nothing is known about their more challenging vertex-connectivity variants. We initiate the study of online vertex-connectivity network design for arbitrary demand pairs by considering the fundamental problem of connectivity augmentation. In an instance of \emph{Online Vertex Connectivity Augmentation} (OVCA), we are given a $k$-vertex-connected graph $F$ and a set of additional purchasable edges (with costs). Demand pairs $(s_1,t_1),(s_2,t_2),\dots$ then arrive online. Upon each arrival, the algorithm must irrevocably purchase edges so that $s_i$ and $t_i$ become $(k{+}1)$-vertex-connected in the union of $F$ with all edges purchased so far. Our main contribution is an $O((k+\log n)\log n)$-competitive online algorithm for OVCA. We also give an offline $O(k+\log n)$-approximation algorithm. We obtain our algorithms via a structural dichotomy on the minimum vertex cuts of $F$. We note that some dependence on $k$ in the approximation ratio is necessary -- i.e., the problem is hard to approximate within any $\mathrm{polylog}(n)$ factor independent of $k$ -- which shows a separation from the edge-connectivity setting. Finally, specializing to $k=1$, our methods yield the first polylogarithmic-competitive online algorithm for a vertex-connectivity survivable network design problem, namely an $\widetilde O(\log^3 n)$-competitive algorithm for online $\{1,2\}$-VC-SNDP (i.e., an SNDP variant where each demand pair comes with a vertex-connectivity requirement of $1$ or $2$).

Thursday, November 12 2026, 01:00

A Parallel Greedy Algorithm for Maximal b-Matching in Hypergraphs

Authors: Saeed Bakhshan, Daniel Grosu, Henrik Reinstädtler, Christian Schulz

We present \parallelBatchBMatch, a parallel batch-vote algorithm for maximal $b$-matching in hypergraphs. The algorithm is deterministic for a given ordering of edges and produces the same output as the sequential greedy algorithm, i.e., the lexicographically first maximal $b$-matching. Under a uniformly random priority order it performs $O(m')$ expected work in the binary fork--join model, where $m'$ is the number of pins, the total number of vertex--edge incidences. The algorithm performs at most $m$ rounds for every priority order, where $m$ is the number of edges. Every round of the algorithm has $O(\log^2 m')$ depth with high probability, and thus the total depth of the algorithm is $O(m \log^2 m')$ with high probability. We implement the parallel algorithm in CUDA and evaluate it on real-world hypergraphs with capacities $b \in \{1, 3, 5\}$. The parallel algorithm computes the same matchings as the greedy algorithms and for the case of $b = 5$ it is up to $19.07$ times faster than the existing state-of-the art greedy algorithm.

Authors: Saeed Bakhshan, Daniel Grosu, Henrik Reinstädtler, Christian Schulz

We present \parallelBatchBMatch, a parallel batch-vote algorithm for maximal $b$-matching in hypergraphs. The algorithm is deterministic for a given ordering of edges and produces the same output as the sequential greedy algorithm, i.e., the lexicographically first maximal $b$-matching. Under a uniformly random priority order it performs $O(m')$ expected work in the binary fork--join model, where $m'$ is the number of pins, the total number of vertex--edge incidences. The algorithm performs at most $m$ rounds for every priority order, where $m$ is the number of edges. Every round of the algorithm has $O(\log^2 m')$ depth with high probability, and thus the total depth of the algorithm is $O(m \log^2 m')$ with high probability. We implement the parallel algorithm in CUDA and evaluate it on real-world hypergraphs with capacities $b \in \{1, 3, 5\}$. The parallel algorithm computes the same matchings as the greedy algorithms and for the case of $b = 5$ it is up to $19.07$ times faster than the existing state-of-the art greedy algorithm.

Thursday, November 12 2026, 01:00

A polynomial-time approximation scheme for the weighted Euclidean Minimum Latency Problem in fixed dimension

Authors: Daniel Skachkov, Yuriy Dorn

In the Traveling Salesman Problem, a route must visit all customers and return to its starting point while minimizing its total length. The Minimum Latency Problem, also known as the Traveling Repairman Problem, instead minimizes the sum of the times at which customers are first visited. We give a deterministic polynomial-time approximation scheme for the weighted Euclidean Minimum Latency Problem in every fixed dimension, with arbitrary nonnegative customer weights. Our proof restricts the form of a solution in three steps, each with a controlled increase in cost. First, a decomposition based on Sitters' method assigns customers to phases that start and end at the root. Second, within each phase, we divide the route into a constant number of blocks and charge each customer at the end of its assigned block. The resulting objective is a nonnegative linear combination of block lengths. Third, we add an auxiliary dimension and use a fixed Minkowski norm to close each block path into a tour. Arora's closed-tour estimate then gives paths that cross the boundaries of a common dissection only at designated portals and a bounded number of times per face. A dynamic program optimizes the restricted objective, and projecting its paths back to the original space gives the final route. Rounding weights and coordinates keeps the state space polynomial.

Authors: Daniel Skachkov, Yuriy Dorn

In the Traveling Salesman Problem, a route must visit all customers and return to its starting point while minimizing its total length. The Minimum Latency Problem, also known as the Traveling Repairman Problem, instead minimizes the sum of the times at which customers are first visited. We give a deterministic polynomial-time approximation scheme for the weighted Euclidean Minimum Latency Problem in every fixed dimension, with arbitrary nonnegative customer weights. Our proof restricts the form of a solution in three steps, each with a controlled increase in cost. First, a decomposition based on Sitters' method assigns customers to phases that start and end at the root. Second, within each phase, we divide the route into a constant number of blocks and charge each customer at the end of its assigned block. The resulting objective is a nonnegative linear combination of block lengths. Third, we add an auxiliary dimension and use a fixed Minkowski norm to close each block path into a tour. Arora's closed-tour estimate then gives paths that cross the boundaries of a common dissection only at designated portals and a bounded number of times per face. A dynamic program optimizes the restricted objective, and projecting its paths back to the original space gives the final route. Rounding weights and coordinates keeps the state space polynomial.

Thursday, November 12 2026, 01:00

Integrality Gap of 3.4 for Santa Claus: A Topology-based Analysis

Authors: Kimon Boehmer, Chien-Chung Huang

In the \textsc{Max-Min Allocation} problem, a set of indivisible items are to be assigned to a set of players and the goal is to maximize the minimum value among all players. A much studied special case, the \textsc{Santa Claus} problem, assumes that an item has a fixed utility but a player is only interested in a subset of items. The main technique for designing approximation algorithms for \textsc{Santa Claus} is the configuration LP, introduced by the Bansal and Sviridenko. A long line of research has progressively improved the upper bound of the integrality gap, mostly relying on the combinatorial arguments of Haxell for finding perfect matchings in certain hypergraphs. Remarkably, a recent paper of Haxell and Szabó (SODA '23) introduced a novel topology-based analysis, by which they are able to obtain the current best upper bound of $3.534$. In this work, we adapt the approach of Haxell and Szabó to analyze the integrality gap. The main new ingredient in our adaptation is that we stop the process of "edge explosions" in a graph used by Haxell and Szabó when it becomes too costly. This early termination of explosions, combined with a new lower bound on a topological connectivity parameter based on a theorem of Aharoni, Berger, and Meshulam, allows us to improve the integrality gap to $3.4$. Our analysis has the additional advantage of being simpler.

Authors: Kimon Boehmer, Chien-Chung Huang

In the \textsc{Max-Min Allocation} problem, a set of indivisible items are to be assigned to a set of players and the goal is to maximize the minimum value among all players. A much studied special case, the \textsc{Santa Claus} problem, assumes that an item has a fixed utility but a player is only interested in a subset of items. The main technique for designing approximation algorithms for \textsc{Santa Claus} is the configuration LP, introduced by the Bansal and Sviridenko. A long line of research has progressively improved the upper bound of the integrality gap, mostly relying on the combinatorial arguments of Haxell for finding perfect matchings in certain hypergraphs. Remarkably, a recent paper of Haxell and Szabó (SODA '23) introduced a novel topology-based analysis, by which they are able to obtain the current best upper bound of $3.534$. In this work, we adapt the approach of Haxell and Szabó to analyze the integrality gap. The main new ingredient in our adaptation is that we stop the process of "edge explosions" in a graph used by Haxell and Szabó when it becomes too costly. This early termination of explosions, combined with a new lower bound on a topological connectivity parameter based on a theorem of Aharoni, Berger, and Meshulam, allows us to improve the integrality gap to $3.4$. Our analysis has the additional advantage of being simpler.

Thursday, November 12 2026, 01:00

Sparse Matrix Multiplication Exponent 1.5 in the Low-bandwidth Model

Authors: Chetan Gupta, Jukka Suomela, Hossein Vahidi

We study sparse matrix multiplication in the low-bandwidth model: $n$ computers multiply two $n\times n$ matrices, and each computer can send and receive one $O(\log n)$-bit message per round. The input matrices are uniformly sparse, meaning that there are at most $d$ nonzero entries in each row and column, and the task is to compute a uniformly sparse part of their product. In prior work, Gupta et al.~(SPAA 2022) presented an $O(d^{1.907})$-round algorithm over fields for this problem when the sparsity structure is globally known in advance (this is referred to as the supported setting). The upper bound was later improved to $O(d^{1.832})$ rounds. In this work, we give a deterministic algorithm that solves this problem in $O(d^{3/2})$ rounds in the supported setting, improving the previous bounds over fields and semirings.

Authors: Chetan Gupta, Jukka Suomela, Hossein Vahidi

We study sparse matrix multiplication in the low-bandwidth model: $n$ computers multiply two $n\times n$ matrices, and each computer can send and receive one $O(\log n)$-bit message per round. The input matrices are uniformly sparse, meaning that there are at most $d$ nonzero entries in each row and column, and the task is to compute a uniformly sparse part of their product. In prior work, Gupta et al.~(SPAA 2022) presented an $O(d^{1.907})$-round algorithm over fields for this problem when the sparsity structure is globally known in advance (this is referred to as the supported setting). The upper bound was later improved to $O(d^{1.832})$ rounds. In this work, we give a deterministic algorithm that solves this problem in $O(d^{3/2})$ rounds in the supported setting, improving the previous bounds over fields and semirings.

Thursday, November 12 2026, 01:00

Speedup-Aware Rounding for Scheduling Malleable Tasks with Precedence Constraints

Authors: Chi-Yeh Chen

We study the scheduling of malleable tasks with precedence constraints on $m$ identical processors to minimize the makespan, assuming that the processing time of a task does not increase and its work does not decrease with the number of processors, and that the work is a convex function of the processing time. The standard two-phase method solves a linear program, rounds the fractional processing times to allotments, caps the allotments at a threshold $μ$, and runs list scheduling. We propose \emph{speedup-aware rounding}, which rounds a task to its slower allotment exactly when this allotment is at most a factor $a$ slower than the fractional processing time. We determine the worst-case increase of the work under this rule exactly and show that no per-task rounding rule with time factor $a$ does better. With $μ=\lceilβ(a)m\rceil$ the algorithm achieves the approximation ratio $13(30+\sqrt{493})/225\approx3.0162$ for every $m$, which improves the previous bound $2+\sqrt2\approx3.4142$; a single task with linear speedup shows that its ratio is at least $\approx2.9558$. For concave speedup functions, which include the roofline, communication, Amdahl and power-law models, the ratio becomes $\approx2.9739$, which improves the ratio $\approx3.2919$ of Jansen and Zhang. We also revisit the strictly decreasing case of our earlier work (Chen and Chu, 2013): the reported ratio $2.9549$ relies on an elimination step that is not valid. In this case the algorithm coincides with that of Jansen and Zhang up to its parameters, and with parameters chosen by the corrected analysis it guarantees their ratio $\approx3.2919$. Finally, without convexity no per-task rounding improves the known time--work trade-off, and the analysis cannot go below $(3+\sqrt5)/2$.

Authors: Chi-Yeh Chen

We study the scheduling of malleable tasks with precedence constraints on $m$ identical processors to minimize the makespan, assuming that the processing time of a task does not increase and its work does not decrease with the number of processors, and that the work is a convex function of the processing time. The standard two-phase method solves a linear program, rounds the fractional processing times to allotments, caps the allotments at a threshold $μ$, and runs list scheduling. We propose \emph{speedup-aware rounding}, which rounds a task to its slower allotment exactly when this allotment is at most a factor $a$ slower than the fractional processing time. We determine the worst-case increase of the work under this rule exactly and show that no per-task rounding rule with time factor $a$ does better. With $μ=\lceilβ(a)m\rceil$ the algorithm achieves the approximation ratio $13(30+\sqrt{493})/225\approx3.0162$ for every $m$, which improves the previous bound $2+\sqrt2\approx3.4142$; a single task with linear speedup shows that its ratio is at least $\approx2.9558$. For concave speedup functions, which include the roofline, communication, Amdahl and power-law models, the ratio becomes $\approx2.9739$, which improves the ratio $\approx3.2919$ of Jansen and Zhang. We also revisit the strictly decreasing case of our earlier work (Chen and Chu, 2013): the reported ratio $2.9549$ relies on an elimination step that is not valid. In this case the algorithm coincides with that of Jansen and Zhang up to its parameters, and with parameters chosen by the corrected analysis it guarantees their ratio $\approx3.2919$. Finally, without convexity no per-task rounding improves the known time--work trade-off, and the analysis cannot go below $(3+\sqrt5)/2$.

Thursday, November 12 2026, 01:00

Parallel Spanning Forest and Biconnectivity for Well-Connected Graph in Sub-Logarithmic Time

Authors: Bojun Liu, S. Cliff Liu, Shuaibo Yang

We give a randomized ARBITRARY CRCW PRAM algorithm for computing a spanning forest of an $n$-vertex, $m$-edge graph. Let $d$ be the maximum component diameter. With good probability, the algorithm runs in $O(\log d+\log\log n)$ time using $O(m+n)$ processors and outputs a rooted spanning forest of original edges with maximum depth $2^{O(\log d+\log\log n)}$. For finding bridges, our algorithm identifies every bridge edge in $O(\log d+\log\log n)$ time. For biconnectivity, it outputs all blocks in $O(\log d+\log d'+\log\log n)$ time, where $d'$ is the bi-diameter. All these problems are fundamental in graph algorithms with numerous applications. Our results improve the best previous time bounds parameterized by the number of vertices and diameter. Finally, we prove deterministic $Ω(\log n/\log\log n)$ lower bounds for outputting a DFS Sequence, a weighted spanning forest, and an ear decomposition.

Authors: Bojun Liu, S. Cliff Liu, Shuaibo Yang

We give a randomized ARBITRARY CRCW PRAM algorithm for computing a spanning forest of an $n$-vertex, $m$-edge graph. Let $d$ be the maximum component diameter. With good probability, the algorithm runs in $O(\log d+\log\log n)$ time using $O(m+n)$ processors and outputs a rooted spanning forest of original edges with maximum depth $2^{O(\log d+\log\log n)}$. For finding bridges, our algorithm identifies every bridge edge in $O(\log d+\log\log n)$ time. For biconnectivity, it outputs all blocks in $O(\log d+\log d'+\log\log n)$ time, where $d'$ is the bi-diameter. All these problems are fundamental in graph algorithms with numerous applications. Our results improve the best previous time bounds parameterized by the number of vertices and diameter. Finally, we prove deterministic $Ω(\log n/\log\log n)$ lower bounds for outputting a DFS Sequence, a weighted spanning forest, and an ear decomposition.

Thursday, November 12 2026, 01:00

Source Assignment: Hardness, Approximation, and Parameterized Algorithms

Authors: Ruxandra Marinescu-Ghemeci, Adrian Miclăuş, Alexandru Popa

In Source Assignment, each pointer represents a substring of a common reference string and may have several possible source occurrences. The goal is to choose one source for each pointer while minimizing the number of distinct reference boundaries used by the selected sources. The problem was introduced by Qiu and Kingsford (Bioinformatics 2021) in the context of compact genome-graph construction. We distinguish \emph{Explicit Source Assignment}, where admissible source sets are given explicitly, from \emph{Complete Source Assignment}, where all occurrences of each pointer string are available. We show that Explicit Source Assignment has an exact value-preserving formulation as cardinality Set Cover with Pairs. For every fixed $\varepsilon>0$, it cannot be approximated within $(1-\varepsilon)\ln p$ unless $\mathrm{P}=\mathrm{NP}$, even over a binary alphabet when all pointers have length one, while known algorithms for Minimum-Weight Multicolored Subgraph give an $O(\sqrt{q\log p})$-approximation for arbitrary pointer lengths. Our main hardness result proves that Complete Source Assignment is APX-hard even over a binary alphabet, with equal pointer lengths and at most two sources per pointer. For edge-disjoint equal-length instances, we give a $3/2$-approximation and a $1.12585$-approximation when each pointer has at most two sources. We also introduce the rainbow-path length $λ$: the problem is polynomial-time solvable for $λ\le2$, and for $λ\le3$ we obtain a $(9/8+\varepsilon)$-approximation. We further give a polynomial-time test for deciding whether abstract source sets can be realized exactly as complete occurrence sets over an unrestricted alphabet. Finally, we give exact fixed-parameter algorithms for several structural parameters, and show that Complete Source Assignment is fixed-parameter tractable in the alphabet size and maximum pointer length.

Authors: Ruxandra Marinescu-Ghemeci, Adrian Miclăuş, Alexandru Popa

In Source Assignment, each pointer represents a substring of a common reference string and may have several possible source occurrences. The goal is to choose one source for each pointer while minimizing the number of distinct reference boundaries used by the selected sources. The problem was introduced by Qiu and Kingsford (Bioinformatics 2021) in the context of compact genome-graph construction. We distinguish \emph{Explicit Source Assignment}, where admissible source sets are given explicitly, from \emph{Complete Source Assignment}, where all occurrences of each pointer string are available. We show that Explicit Source Assignment has an exact value-preserving formulation as cardinality Set Cover with Pairs. For every fixed $\varepsilon>0$, it cannot be approximated within $(1-\varepsilon)\ln p$ unless $\mathrm{P}=\mathrm{NP}$, even over a binary alphabet when all pointers have length one, while known algorithms for Minimum-Weight Multicolored Subgraph give an $O(\sqrt{q\log p})$-approximation for arbitrary pointer lengths. Our main hardness result proves that Complete Source Assignment is APX-hard even over a binary alphabet, with equal pointer lengths and at most two sources per pointer. For edge-disjoint equal-length instances, we give a $3/2$-approximation and a $1.12585$-approximation when each pointer has at most two sources. We also introduce the rainbow-path length $λ$: the problem is polynomial-time solvable for $λ\le2$, and for $λ\le3$ we obtain a $(9/8+\varepsilon)$-approximation. We further give a polynomial-time test for deciding whether abstract source sets can be realized exactly as complete occurrence sets over an unrestricted alphabet. Finally, we give exact fixed-parameter algorithms for several structural parameters, and show that Complete Source Assignment is fixed-parameter tractable in the alphabet size and maximum pointer length.

Thursday, November 12 2026, 01:00

A Sublogarithmic Competitive Algorithm for Online Graph Exploration

Authors: Max Klimm

In online graph exploration, an agent must visit every vertex of an unknown weighted graph and return to its start, learning the incident edges at each vertex it visits. On $n$-vertex graphs, the best known competitive ratio has been $O(\log n)$, achieved by several algorithms, among them the hierarchical depth-first search of Megow, Mehlhorn, and Schweitzer. We give an algorithm with competitive ratio ${(2+o(1))}\ln n/\ln\ln n$ for known~$n$. Like hierarchical depth-first search, it groups the edges into geometric weight classes and searches depth-first between components of lighter edges. Unlike it, the agent tours each component along a minimum spanning tree for the original weights, so only crossings between components pay for rounding, and the base $b$ of the weight class can grow with~$n$. The leading constant $2$ is tight for our choice of base. Without knowing $n$, restarts with increasing size guesses give upper bounds with leading constant $4$ for a fixed schedule and $2$ in expectation for a randomly shifted schedule. If the weights lie in a known range $[1,2^k]$ with $k\ge4$, choosing $b=k/\log_2 k$ gives a competitive ratio of $O(k/\log k)$ without knowing $n$.

Authors: Max Klimm

In online graph exploration, an agent must visit every vertex of an unknown weighted graph and return to its start, learning the incident edges at each vertex it visits. On $n$-vertex graphs, the best known competitive ratio has been $O(\log n)$, achieved by several algorithms, among them the hierarchical depth-first search of Megow, Mehlhorn, and Schweitzer. We give an algorithm with competitive ratio ${(2+o(1))}\ln n/\ln\ln n$ for known~$n$. Like hierarchical depth-first search, it groups the edges into geometric weight classes and searches depth-first between components of lighter edges. Unlike it, the agent tours each component along a minimum spanning tree for the original weights, so only crossings between components pay for rounding, and the base $b$ of the weight class can grow with~$n$. The leading constant $2$ is tight for our choice of base. Without knowing $n$, restarts with increasing size guesses give upper bounds with leading constant $4$ for a fixed schedule and $2$ in expectation for a randomly shifted schedule. If the weights lie in a known range $[1,2^k]$ with $k\ge4$, choosing $b=k/\log_2 k$ gives a competitive ratio of $O(k/\log k)$ without knowing $n$.

Thursday, November 12 2026, 01:00

Multiple-Choice Secretary Problem with Predictions

Authors: Jing Yuan, Qi Cai, Xin Gao, Shaojie Tang

We study the multiple-choice secretary problem, in which candidates arrive in uniformly random order and the goal is to select at most $k$ candidates to maximize their total value. We consider a setting where predictions of individual candidate values are available before any candidate arrives. Fujii and Yoshida's Learned Kleinberg algorithm achieves competitive loss $O(\min\{\log k/\sqrt{k},\eps\})$, where $k$ is the selection capacity and $\eps$ is the maximum multiplicative prediction error. We remove the logarithmic factor through a sharper analysis of the same switching algorithm with a retuned threshold. The resulting competitive ratio is at least $\max\{0,1-\min\{32/\sqrt{k},4\eps\}\}$. In particular, exact predictions ($\eps=0$) achieve the offline optimum on every arrival order.

Authors: Jing Yuan, Qi Cai, Xin Gao, Shaojie Tang

We study the multiple-choice secretary problem, in which candidates arrive in uniformly random order and the goal is to select at most $k$ candidates to maximize their total value. We consider a setting where predictions of individual candidate values are available before any candidate arrives. Fujii and Yoshida's Learned Kleinberg algorithm achieves competitive loss $O(\min\{\log k/\sqrt{k},\eps\})$, where $k$ is the selection capacity and $\eps$ is the maximum multiplicative prediction error. We remove the logarithmic factor through a sharper analysis of the same switching algorithm with a retuned threshold. The resulting competitive ratio is at least $\max\{0,1-\min\{32/\sqrt{k},4\eps\}\}$. In particular, exact predictions ($\eps=0$) achieve the offline optimum on every arrival order.

Thursday, November 12 2026, 01:00

Fall Jobs Post 2026

from Computational Complexity

Each fall I try to predict the upcoming job market for theoretical computer science faculty and give suggestions to those pursuing these positions. Last year I wrote that we should get set for a rocky ride, with AI’s disruption of computer science, fiscal stress at universities, and new U.S. policies affecting visas and funding.

Rocky indeed. The job market for computer science graduates continues to decline and with it undergraduate enrollment in the discipline. That should crater demand for CS professors, but anecdotally I'm seeing quite a few departments advertising multiple slots. Perhaps as more computer science departments create Artificial Intelligence programs, deans see the need for continued growth in computing.

Mostly though we're seeing hiring at schools with reasonably good finances, those with large endowments and state flagship universities. Many other schools are struggling and having hiring cutbacks or outright freezes. So I expect the strongest theory students will still find good positions but the second tier might be more challenging than in the past. Oddly enough, a candidate may have an easier time landing a job at a top university than at a mid-tier one.

I'm pessimistic that the recent elections will change much, at least for the next couple of years. Democrats won't make higher education a priority issue. At best they will slow the development of AI but it will continue to disrupt not just computer science but all of higher ed. Trump will remain president for two more years so I don't expect we'll see major changes in visa policy, at least not in a positive direction. Other causes of financial stress, including Baumol's cost disease, the tough job market for new computer science graduates, the public perception of higher education, and the demographic cliff will not go away. Perhaps universities will learn how to use artificial intelligence to cut costs but I'm not holding my breath.

On the positive side, Congress could use the appropriations process to reopen the NSF spigot, though that could take years to play out.

On to specific advice for students on the academic job market.

CATCS maintains a Theoretical Computer Science Jobs posting site. More generally in computer science, there is the CRA Database of Job Candidates and the CRA and ACM job postings.

Don't shy away from artificial intelligence. Make sure you talk about AI in both your research and teaching statements. Departments will need instructors who can teach the courses in all the new AI majors popping up.

Have a well-designed website with all your job materials and links to your papers (AI can help). Make sure your Google Scholar, LinkedIn and GitHub (if relevant) sites are accurate and up to date. Make a short video describing your research to a general computer science crowd. Take advantage of all the links above. Reach out directly to professors you know at schools you are interested in. And start now.

By Lance Fortnow

Each fall I try to predict the upcoming job market for theoretical computer science faculty and give suggestions to those pursuing these positions. Last year I wrote that we should get set for a rocky ride, with AI’s disruption of computer science, fiscal stress at universities, and new U.S. policies affecting visas and funding.

Rocky indeed. The job market for computer science graduates continues to decline and with it undergraduate enrollment in the discipline. That should crater demand for CS professors, but anecdotally I'm seeing quite a few departments advertising multiple slots. Perhaps as more computer science departments create Artificial Intelligence programs, deans see the need for continued growth in computing.

Mostly though we're seeing hiring at schools with reasonably good finances, those with large endowments and state flagship universities. Many other schools are struggling and having hiring cutbacks or outright freezes. So I expect the strongest theory students will still find good positions but the second tier might be more challenging than in the past. Oddly enough, a candidate may have an easier time landing a job at a top university than at a mid-tier one.

I'm pessimistic that the recent elections will change much, at least for the next couple of years. Democrats won't make higher education a priority issue. At best they will slow the development of AI but it will continue to disrupt not just computer science but all of higher ed. Trump will remain president for two more years so I don't expect we'll see major changes in visa policy, at least not in a positive direction. Other causes of financial stress, including Baumol's cost disease, the tough job market for new computer science graduates, the public perception of higher education, and the demographic cliff will not go away. Perhaps universities will learn how to use artificial intelligence to cut costs but I'm not holding my breath.

On the positive side, Congress could use the appropriations process to reopen the NSF spigot, though that could take years to play out.

On to specific advice for students on the academic job market.

CATCS maintains a Theoretical Computer Science Jobs posting site. More generally in computer science, there is the CRA Database of Job Candidates and the CRA and ACM job postings.

Don't shy away from artificial intelligence. Make sure you talk about AI in both your research and teaching statements. Departments will need instructors who can teach the courses in all the new AI majors popping up.

Have a well-designed website with all your job materials and links to your papers (AI can help). Make sure your Google Scholar, LinkedIn and GitHub (if relevant) sites are accurate and up to date. Make a short video describing your research to a general computer science crowd. Take advantage of all the links above. Reach out directly to professors you know at schools you are interested in. And start now.

By Lance Fortnow

Wednesday, November 11 2026, 16:38

TR26-317 | Explicit Subspace Evasive Sets via Subspace Designs | Yeyuan Chen, Joshua Brakensiek, Rohan Goyal, Venkatesan Guruswami, Mahdi Cheraghchi, Aaron (Louie) Putterman, Zihan Zhang

from ECCC Papers

Given a finite field $\mathbb F_q$, we say that a set $A\subseteq \mathbb F^n_q$ is an $(r,c)$ subspace evasive set if and only if for any affine subspace $U\subseteq \mathbb F^n_q$ with bounded dimension $\dim(U)\leq r$, the intersection size $|A\cap U|$ is at most $c$. The probabilistic method shows the existence of $(r,c)$ subspace evasive sets with $|A| = q^{(1-\epsilon) n}$ with $c = O(r/\epsilon)$. However, the best explicit constructions by Dvir--Lovett (STOC 2012) and Ben-Aroya--Shinkar (CJTCS 2013) have $c$ growing as an exponential function of $r$. Here, we give an explicit construction of such subspace evasive set that achieves the intersection bound $c=O(\min(r\log(r/\epsilon)/\epsilon, r^2/\epsilon))$ when both $r$ and $1/\epsilon$ are constants. Our approach is simple and the key idea is to compose subspace design codes (Guruswami-Xing, STOC 2013) with a suitably chosen ``local'' subspace evasive set. As a corollary, we give the first construction of an explicit (strongly) capacity-achieving list-recoverable code with nearly optimal list size, circumventing all previously-known barriers to construction.

Wednesday, November 11 2026, 09:23

Solomonoff Induction with Reflective Oracles

Authors: Damini Kusum

We study predictive properties of a reflective variant of Solomonoff induction formulated using reflective oracles (Fallenstein et al. 2015, Wyeth et al. 2025). We show that the resulting optimal predictor succeeds against computable predictive strategies that specify only part of the data-generating process. The argument relies on the fact that hypotheses computable from the predictor's own conditional probabilities remain within the reflective hypothesis class. In particular, the predictor learns incomplete models specified by computable interval restrictions on next-bit probabilities. We also establish weighted forms of unbiasedness and calibration. Finally, we discuss whether an optimal predictor's conditional probabilities suffice to compute every hypothesis in the reflective class.

Authors: Damini Kusum

We study predictive properties of a reflective variant of Solomonoff induction formulated using reflective oracles (Fallenstein et al. 2015, Wyeth et al. 2025). We show that the resulting optimal predictor succeeds against computable predictive strategies that specify only part of the data-generating process. The argument relies on the fact that hypotheses computable from the predictor's own conditional probabilities remain within the reflective hypothesis class. In particular, the predictor learns incomplete models specified by computable interval restrictions on next-bit probabilities. We also establish weighted forms of unbiasedness and calibration. Finally, we discuss whether an optimal predictor's conditional probabilities suffice to compute every hypothesis in the reflective class.

Wednesday, November 11 2026, 01:00

Uniform Quantum Average-Case Fine-Grained Hardness of Arithmetized $k$-Orthogonal Vectors

Authors: Dhruv Prasad Deshpande, Abhishek Mishra

We study exact evaluation of the arithmetized $k$-Orthogonal Vectors polynomial $FOV^k$ of Ball et al. (2017, 2018). For every fixed integer $k \geq 3$, set $d(n) = \left \lceil \log^2 n \right \rceil$. Assuming the basic decision version of Quantum Strong Exponential-Time Hypothesis (QSETH), for every constant $\varepsilon > 0$, no uniform quantum algorithm runs in $O(n^{k/2 - \varepsilon})$ time on every input and evaluates this polynomial correctly with probability at least $3 / 4$ over a uniformly random input in $\mathbb F_{p(n)}^{knd(n)}$ and the algorithm's internal randomness, for all sufficiently large $n$. This holds when $p(n)$ is the smallest prime greater than $n^{k - 1}$ and when it is the smallest prime greater than $n^k$. Our direct reduction combines classical polynomial self-correction with calls to a quantum average-case evaluator. For the smaller prime, random sampling finds an orthogonal tuple with constant probability whenever the count is large enough to wrap around modulo $p(n)$; otherwise, the exact residue determines the answer. For the larger prime, quantum minimum finding and an asymptotic prime-gap bound account for the cost of generating the prescribed field. In view of prior isolation-based reductions permitting small characteristics, we present these results as explicit reductions for the two canonical-prime families, with all setup costs included in the uniform quantum running time. The hardness guarantee concerns single-instance exact evaluation; a matching quantum evaluation algorithm and a quantum proof-of-work protocol remain open.

Authors: Dhruv Prasad Deshpande, Abhishek Mishra

We study exact evaluation of the arithmetized $k$-Orthogonal Vectors polynomial $FOV^k$ of Ball et al. (2017, 2018). For every fixed integer $k \geq 3$, set $d(n) = \left \lceil \log^2 n \right \rceil$. Assuming the basic decision version of Quantum Strong Exponential-Time Hypothesis (QSETH), for every constant $\varepsilon > 0$, no uniform quantum algorithm runs in $O(n^{k/2 - \varepsilon})$ time on every input and evaluates this polynomial correctly with probability at least $3 / 4$ over a uniformly random input in $\mathbb F_{p(n)}^{knd(n)}$ and the algorithm's internal randomness, for all sufficiently large $n$. This holds when $p(n)$ is the smallest prime greater than $n^{k - 1}$ and when it is the smallest prime greater than $n^k$. Our direct reduction combines classical polynomial self-correction with calls to a quantum average-case evaluator. For the smaller prime, random sampling finds an orthogonal tuple with constant probability whenever the count is large enough to wrap around modulo $p(n)$; otherwise, the exact residue determines the answer. For the larger prime, quantum minimum finding and an asymptotic prime-gap bound account for the cost of generating the prescribed field. In view of prior isolation-based reductions permitting small characteristics, we present these results as explicit reductions for the two canonical-prime families, with all setup costs included in the uniform quantum running time. The hardness guarantee concerns single-instance exact evaluation; a matching quantum evaluation algorithm and a quantum proof-of-work protocol remain open.

Wednesday, November 11 2026, 01:00

Complexity of Deciding Existence of Maltsev Polymorphisms of Finite Relational Structures

Authors: Dejan Delic, Ali Syed

In this article, we show that the problem of deciding whether a finite relational structure $\mathbf{A}$ admits a Maltsev polymorphism is a computational problem, which can be solved in polynomial time. We provide a proof of this result by reducing the problem to deciding whether a binary relational structure, associated with $\mathbf{A}$ admits such a polymorphism. In our construction of such an algorithm, we make an extensive use of a property of binary relations, which is implied by the existence of a Maltsev polymorphism, the so-called rectangularity property.

Authors: Dejan Delic, Ali Syed

In this article, we show that the problem of deciding whether a finite relational structure $\mathbf{A}$ admits a Maltsev polymorphism is a computational problem, which can be solved in polynomial time. We provide a proof of this result by reducing the problem to deciding whether a binary relational structure, associated with $\mathbf{A}$ admits such a polymorphism. In our construction of such an algorithm, we make an extensive use of a property of binary relations, which is implied by the existence of a Maltsev polymorphism, the so-called rectangularity property.

Wednesday, November 11 2026, 01:00

$\mathsf{SVP}_\infty$ Is NP-Hard for Power-of-two Cyclotomic Ideals

Authors: Yansong Feng, Jiaqi Liu, Hengyi Luo, Yanbin Pan

We prove that exact decision-$\mathsf{SVP}$ in the coefficient $\ell_\infty$-norm is $\mathsf{NP}$-complete for principal ideals and free modules of rank two over the cyclotomic rings $R_n=\mathbb{Z}[X]/(X^n+1)$, where $n$ ranges over powers of two. For each case, we give a deterministic polynomial-time many-one reduction from the Subset Sum problem. For an instance with $s$ positive input integers, the output lattice has dimension $O(s)$. A solution to the original Subset Sum instance can be recovered in polynomial time from any nonzero output lattice vector of norm at most the specified bound. We also transfer these reductions from $R_n$ to the power-of-two cyclic rings $S_{2n}=\mathbb{Z}[X]/(X^{2n}-1)$. This gives $\mathsf{NP}$-completeness of decision-$\mathsf{SVP}$ in the coefficient $\ell_\infty$-norm for principal ideals and free modules of any fixed positive rank over these rings.

Authors: Yansong Feng, Jiaqi Liu, Hengyi Luo, Yanbin Pan

We prove that exact decision-$\mathsf{SVP}$ in the coefficient $\ell_\infty$-norm is $\mathsf{NP}$-complete for principal ideals and free modules of rank two over the cyclotomic rings $R_n=\mathbb{Z}[X]/(X^n+1)$, where $n$ ranges over powers of two. For each case, we give a deterministic polynomial-time many-one reduction from the Subset Sum problem. For an instance with $s$ positive input integers, the output lattice has dimension $O(s)$. A solution to the original Subset Sum instance can be recovered in polynomial time from any nonzero output lattice vector of norm at most the specified bound. We also transfer these reductions from $R_n$ to the power-of-two cyclic rings $S_{2n}=\mathbb{Z}[X]/(X^{2n}-1)$. This gives $\mathsf{NP}$-completeness of decision-$\mathsf{SVP}$ in the coefficient $\ell_\infty$-norm for principal ideals and free modules of any fixed positive rank over these rings.

Wednesday, November 11 2026, 01:00

From Worst-Case to Smoothed Complexity in Continuous Optimization

Authors: Daniel Cortild, Coralia Cartis

Worst-case complexity bounds often fail to reflect the practical performance of optimization algorithms, while average-case complexity heavily relies on the distribution of the input. In this work, we develop a smoothed-analysis framework for classical continuous optimization algorithms by studying their behaviour on random perturbations of arbitrary functions. Our main contribution is to apply methods suited for non-degenerate functions to perturbations of degenerate functions, thereby recovering the faster best-known rates for the non-degenerate class. We derive smoothed complexity guarantees for several standard methods, including (Accelerated) Gradient Descent, Conjugate Gradient, and Newton-type methods, under various settings including nonconvex ones, and show that these bounds can substantially improve upon their worst-case guarantees, bringing them closer to the practical behaviour of the respective methods.

Authors: Daniel Cortild, Coralia Cartis

Worst-case complexity bounds often fail to reflect the practical performance of optimization algorithms, while average-case complexity heavily relies on the distribution of the input. In this work, we develop a smoothed-analysis framework for classical continuous optimization algorithms by studying their behaviour on random perturbations of arbitrary functions. Our main contribution is to apply methods suited for non-degenerate functions to perturbations of degenerate functions, thereby recovering the faster best-known rates for the non-degenerate class. We derive smoothed complexity guarantees for several standard methods, including (Accelerated) Gradient Descent, Conjugate Gradient, and Newton-type methods, under various settings including nonconvex ones, and show that these bounds can substantially improve upon their worst-case guarantees, bringing them closer to the practical behaviour of the respective methods.

Wednesday, November 11 2026, 01:00

The Monophonic Directed Convexities

Authors: Júlio Araújo, Ana Karolinna Maia, Thiago Marcilon, Nicolas Nisse, Rudini M. Sampaio

Graph Convexity is a topic with many papers in the literature, especially on the $P_3$, geodesic, and monophonic convexities in undirected graphs. Regarding directed graphs, Erd{\H o}s et al. in 1972 and Chartrand and Zhang in 2000 obtained the first results on the $P_3$ and geodesic convexities for directed graphs, respectively. In this paper, we start the investigation of monophonic convexity on directed graphs. We define two convexities, the monophonic and the strong monophonic directed convexities, related to the minimal and the induced directed paths, respectively. We show that deciding whether a set is convex is polynomial-time solvable in the monophonic convexity, but coNP-complete in the strong monophonic convexity. For both convexities, we prove that the hull number is NP-complete even in DAGs. Regarding the monophonic convexity, we prove that the hull number is polynomial time solvable in tournaments, but the interval number is NP-complete in tournaments and also in bipartite oriented graphs.

Authors: Júlio Araújo, Ana Karolinna Maia, Thiago Marcilon, Nicolas Nisse, Rudini M. Sampaio

Graph Convexity is a topic with many papers in the literature, especially on the $P_3$, geodesic, and monophonic convexities in undirected graphs. Regarding directed graphs, Erd{\H o}s et al. in 1972 and Chartrand and Zhang in 2000 obtained the first results on the $P_3$ and geodesic convexities for directed graphs, respectively. In this paper, we start the investigation of monophonic convexity on directed graphs. We define two convexities, the monophonic and the strong monophonic directed convexities, related to the minimal and the induced directed paths, respectively. We show that deciding whether a set is convex is polynomial-time solvable in the monophonic convexity, but coNP-complete in the strong monophonic convexity. For both convexities, we prove that the hull number is NP-complete even in DAGs. Regarding the monophonic convexity, we prove that the hull number is polynomial time solvable in tournaments, but the interval number is NP-complete in tournaments and also in bipartite oriented graphs.

Wednesday, November 11 2026, 01:00

Feedback Vertex Set, Max Cut, and Edge Coloring on the Strong Product of a Graph and a Path

Authors: Jan Diester, Kolja Kühn, Manuel Rapp, Samuel Schneider, Torsten Ueckerdt

We examine the computational complexity of well-known problems on subgraphs of the strong product of two paths. Namely, for Max Cut we show NP-hardness on induced subgraphs of $P \boxtimes P$ answering a question by Dvořák, Huynh, Joret, Liu, and Wood (2021). For 3-Edge Coloring we show NP-hardness on subgraphs of $P \boxtimes P$ and that it is equivalent to 3-Edge Coloring on planar graphs for induced subgraphs of $P \boxtimes P$. It is a long standing open question whether 3-Edge Coloring on planar graphs is NP-hard. As a second contribution, we consider graphs of the form $G \boxtimes P$ and show that Feedback Vertex Set on these graphs is FPT in the clique-width of $G$. Furthermore, we show that it can be solved in polynomial time if $G$ is chordal or co-comparable and that the computational complexity depends on the parity of $P$ if $G$ is bipartite. Namely, it is solvable in polynomial time if $P$ has an even number of vertices but is NP-complete otherwise. For Max Cut, we show that it can be solved in polynomial time if $G$ is bipartite.

Authors: Jan Diester, Kolja Kühn, Manuel Rapp, Samuel Schneider, Torsten Ueckerdt

We examine the computational complexity of well-known problems on subgraphs of the strong product of two paths. Namely, for Max Cut we show NP-hardness on induced subgraphs of $P \boxtimes P$ answering a question by Dvořák, Huynh, Joret, Liu, and Wood (2021). For 3-Edge Coloring we show NP-hardness on subgraphs of $P \boxtimes P$ and that it is equivalent to 3-Edge Coloring on planar graphs for induced subgraphs of $P \boxtimes P$. It is a long standing open question whether 3-Edge Coloring on planar graphs is NP-hard. As a second contribution, we consider graphs of the form $G \boxtimes P$ and show that Feedback Vertex Set on these graphs is FPT in the clique-width of $G$. Furthermore, we show that it can be solved in polynomial time if $G$ is chordal or co-comparable and that the computational complexity depends on the parity of $P$ if $G$ is bipartite. Namely, it is solvable in polynomial time if $P$ has an even number of vertices but is NP-complete otherwise. For Max Cut, we show that it can be solved in polynomial time if $G$ is bipartite.

Wednesday, November 11 2026, 01:00

Local Modifications of Nearest-Neighbor and Min-Plus Representations

Authors: Liliia Redina

We study how changing a Boolean-valued function at $d$ inputs affects the size and precision of its exact representations. Starting from a representation of size $m$, we prove four bounds, two for nearest-neighbor (NN) representations and two for integer min-plus representations. (i) For NN representations on a finite subset of $\mathbb{R}^n$ with $n\ge2$, at most $m+2d$ anchors suffice. The coefficient $2$ is optimal: on a three-point domain, changing one value can increase the minimum number of anchors from one to three. (ii) For NN representations on the Boolean cube with rational anchors, let $R$ be the maximum original coordinate bit length, and let $Δ$ be the maximum bit length of the least common denominator of any one original anchor's coordinates. We construct a representation with at most $m+3d$ anchors on a common dyadic grid and coordinate bit length $O(R+Δ+\log(n+1))$. (iii) For integer min-plus representations on the Boolean cube, we give a construction that keeps the original $m$ forms and their labels and adds at most $d$ new forms. Its maximum weight is at most $W+2n$, where $W$ is the largest absolute original coefficient, including constant terms. (iv) For integer min-plus representations on a finite rational domain $X\subset\mathbb{Q}^n$, at most $m+2d$ forms suffice. Their maximum weight is polynomially bounded in $m,d,W,n,|X|$, the coordinate bound, and the common denominator of $X$. The coefficient $2$ is again optimal: on a three-point domain, changing one value can increase the minimum number of forms from one to three. These bounds imply stability under polynomially many changes on the cube for polynomial-size representations, including those with polynomially bounded NN coordinate bit length or min-plus weight.

Authors: Liliia Redina

We study how changing a Boolean-valued function at $d$ inputs affects the size and precision of its exact representations. Starting from a representation of size $m$, we prove four bounds, two for nearest-neighbor (NN) representations and two for integer min-plus representations. (i) For NN representations on a finite subset of $\mathbb{R}^n$ with $n\ge2$, at most $m+2d$ anchors suffice. The coefficient $2$ is optimal: on a three-point domain, changing one value can increase the minimum number of anchors from one to three. (ii) For NN representations on the Boolean cube with rational anchors, let $R$ be the maximum original coordinate bit length, and let $Δ$ be the maximum bit length of the least common denominator of any one original anchor's coordinates. We construct a representation with at most $m+3d$ anchors on a common dyadic grid and coordinate bit length $O(R+Δ+\log(n+1))$. (iii) For integer min-plus representations on the Boolean cube, we give a construction that keeps the original $m$ forms and their labels and adds at most $d$ new forms. Its maximum weight is at most $W+2n$, where $W$ is the largest absolute original coefficient, including constant terms. (iv) For integer min-plus representations on a finite rational domain $X\subset\mathbb{Q}^n$, at most $m+2d$ forms suffice. Their maximum weight is polynomially bounded in $m,d,W,n,|X|$, the coordinate bound, and the common denominator of $X$. The coefficient $2$ is again optimal: on a three-point domain, changing one value can increase the minimum number of forms from one to three. These bounds imply stability under polynomially many changes on the cube for polynomial-size representations, including those with polynomially bounded NN coordinate bit length or min-plus weight.

Wednesday, November 11 2026, 01:00

Computational Power of Clifford and Ising Quantum Computers

Authors: Sawyer Dobson, Zhenghan Wang

We extend the efficient classical simulation of Ising quantum computers from braidings to measurements. We investigate several different definitions of efficient Clifford quantum computation by varying the power of the circuit generator. In case the circuit generator can see the entire input, we give a detailed proof that efficient simulation of classical computation would imply the separation of complexity classes $\mathbf{P}$ and $\mathbf{PSPACE}$. If the circuit generator sees only the input length, we prove that Clifford quantum computers decide what we call affine languages.

Authors: Sawyer Dobson, Zhenghan Wang

We extend the efficient classical simulation of Ising quantum computers from braidings to measurements. We investigate several different definitions of efficient Clifford quantum computation by varying the power of the circuit generator. In case the circuit generator can see the entire input, we give a detailed proof that efficient simulation of classical computation would imply the separation of complexity classes $\mathbf{P}$ and $\mathbf{PSPACE}$. If the circuit generator sees only the input length, we prove that Clifford quantum computers decide what we call affine languages.

Wednesday, November 11 2026, 01:00

Constant query LCCs over the reals

Authors: Zeev Dvir

This note describes a new construction, found by ChatGPT Astra, of constant query Locally Correctable Codes (LCCs) over the real numbers. Prior constructions of constant query LCCs were exclusive to fields of positive characteristic and were conjectured not to exist over the reals. The construction uses low degree polynomial evaluations over an integer grid, together with a novel sampling procedure which outputs, for a given grid point $P$, a random quadratic curve through $P$ which intersects the grid in many other points (the decoding queries). The sampler is `smooth' in the sense that no grid point is queried with high probability. These codes have the additional property that the queried locations (the curve/grid intersections) are all nearby grid points.

Authors: Zeev Dvir

This note describes a new construction, found by ChatGPT Astra, of constant query Locally Correctable Codes (LCCs) over the real numbers. Prior constructions of constant query LCCs were exclusive to fields of positive characteristic and were conjectured not to exist over the reals. The construction uses low degree polynomial evaluations over an integer grid, together with a novel sampling procedure which outputs, for a given grid point $P$, a random quadratic curve through $P$ which intersects the grid in many other points (the decoding queries). The sampler is `smooth' in the sense that no grid point is queried with high probability. These codes have the additional property that the queried locations (the curve/grid intersections) are all nearby grid points.

Wednesday, November 11 2026, 01:00

Linear-size graphs at every strict dismantling level

Authors: Jesús F. Espinoza, Martín Eduardo Frías-Armenta, Rosalía G. Hernández-Amador, Héctor A. Hernández-Hernández

Dismantling reduces a graph to one vertex by successive deletions. At level zero, a vertex is eligible if one of its neighbours is adjacent to all its other neighbours. At level $k\geq1$, its neighbours must induce a graph dismantlable at level $k-1$. A graph has strict level $k$ if it admits a reduction at level $k$ and none at level $k-1$. For every $k\geq1$, we construct a graph of strict level $k$ with $3k+8$ vertices. Each step adds three vertices: the neighbours of one induce the preceding graph, and the other two provide the adjacencies used to rule out other first deletions at the lower level. A second construction gives a graph of order $4k+4$ at strict level $k$. Together the two families give the upper bound $\min\{4k+4,3k+8\}$ on the least order at strict level $k$.

Authors: Jesús F. Espinoza, Martín Eduardo Frías-Armenta, Rosalía G. Hernández-Amador, Héctor A. Hernández-Hernández

Dismantling reduces a graph to one vertex by successive deletions. At level zero, a vertex is eligible if one of its neighbours is adjacent to all its other neighbours. At level $k\geq1$, its neighbours must induce a graph dismantlable at level $k-1$. A graph has strict level $k$ if it admits a reduction at level $k$ and none at level $k-1$. For every $k\geq1$, we construct a graph of strict level $k$ with $3k+8$ vertices. Each step adds three vertices: the neighbours of one induce the preceding graph, and the other two provide the adjacencies used to rule out other first deletions at the lower level. A second construction gives a graph of order $4k+4$ at strict level $k$. Together the two families give the upper bound $\min\{4k+4,3k+8\}$ on the least order at strict level $k$.

Wednesday, November 11 2026, 01:00

Unilluminable rooms: minimal examples, classifications and exceptions

Authors: J. C. F. de Winter, George W. Tokarsky

A room with mirrored walls can contain two points that are mutually invisible: no light ray from one reaches the other. Tokarsky built the first polygonal examples by tiling a room with reflected copies of a triangle and making every tile vertex of certain kinds a corner of the room. We study the smallest rooms of this kind, which we call lattice rooms. Their interior tile vertices of the remaining kind, the sources, are invisible and pairwise mutually invisible. On the tiling by isosceles right triangles, a hole-free lattice room with a source has at least 12 sides, attained only by the pinwheel. On the square tiling the least number is 16, again attained by one room. A hole-free lattice room with two sources has at least 24 sides on the first tiling and 34 on the square tiling. Up to congruence there are 12 such rooms with 24 sides, 89 with 25 sides and 14 with 34 sides. One absorbing post, one partition, or one hole that touches the outer boundary at one point lowers the least number of sides for two sources to 18, 20 and 22 (21 if sides are counted as maximal straight segments). Two and three posts give 17 and 16 sides, and a hole that does not touch the outer boundary gives 26. We list all smallest rooms. Rooms assembled from n modules have exactly 12n sides and n sources, and a hole-free lattice room with three or more sources has at least 36 sides. We also give rooms in which some tile vertices need not be corners: 15 types of 27-sided rooms on the tiling by triangles with angles 30, 60 and 90 degrees, which are smallest in their class, a nonagon with three mutually invisible corners, and a 42-sided room in which a rotation through 180 degrees takes the place of a corner. In a ring of three trapezoids a light ray returns to a corner that is invisible in one trapezoid. Several classifications rely on exhaustive computer searches made finite by the bounds proved here.

Authors: J. C. F. de Winter, George W. Tokarsky

A room with mirrored walls can contain two points that are mutually invisible: no light ray from one reaches the other. Tokarsky built the first polygonal examples by tiling a room with reflected copies of a triangle and making every tile vertex of certain kinds a corner of the room. We study the smallest rooms of this kind, which we call lattice rooms. Their interior tile vertices of the remaining kind, the sources, are invisible and pairwise mutually invisible. On the tiling by isosceles right triangles, a hole-free lattice room with a source has at least 12 sides, attained only by the pinwheel. On the square tiling the least number is 16, again attained by one room. A hole-free lattice room with two sources has at least 24 sides on the first tiling and 34 on the square tiling. Up to congruence there are 12 such rooms with 24 sides, 89 with 25 sides and 14 with 34 sides. One absorbing post, one partition, or one hole that touches the outer boundary at one point lowers the least number of sides for two sources to 18, 20 and 22 (21 if sides are counted as maximal straight segments). Two and three posts give 17 and 16 sides, and a hole that does not touch the outer boundary gives 26. We list all smallest rooms. Rooms assembled from n modules have exactly 12n sides and n sources, and a hole-free lattice room with three or more sources has at least 36 sides. We also give rooms in which some tile vertices need not be corners: 15 types of 27-sided rooms on the tiling by triangles with angles 30, 60 and 90 degrees, which are smallest in their class, a nonagon with three mutually invisible corners, and a 42-sided room in which a rotation through 180 degrees takes the place of a corner. In a ring of three trapezoids a light ray returns to a corner that is invisible in one trapezoid. Several classifications rely on exhaustive computer searches made finite by the bounds proved here.

Wednesday, November 11 2026, 01:00

Parameterized Complexity and Geometric Approximation of Riesz Energy Subset Selection

Authors: Michael Emmerich, Soumen Atta

We choose $k$ of $n$ sites to minimize Riesz $s$-energy, in which a pair at distance $t$ contributes $t^{-s}$. For $s=2$, exact selection is polynomial on a line but NP-hard in the plane. Adapting the proxy lemma of Ceccarello et al. gives an efficient parameterized approximation scheme (EPAS) for fixed Euclidean dimension and fixed $s>0$. To sharpen the bound, we start with a geometric packing. Farthest-first traversal gives $2k$ centers separated by at least the last insertion distance $r$. If one site of an optimum is removed, each of the $k-1$ remaining sites excludes at most one center closer than $r/2$, leaving at least $k+1$ centers free. The new step is a discrete exchange bound: the removed site's interaction energy is at most the average it would contribute at those free centers. In the plane at $s=2$, this reduces the representative bound from $O(k^2\log(2k)/\varepsilon^2)$ for a packing estimate of total energy to $O(k\log(2k)/\varepsilon^2)$, independently of $n$. Rational planar examples show that the resulting separation scale $r/\sqrt{\log(2k)}$ is sharp. A $(1+\varepsilon)$ approximation on rational planar sites takes $O(nk+n\log n)+(C\log(2k)/\varepsilon^2)^k$ operations of polynomial bit cost. The observed number $M$ of representatives determines the search size $\binom{M}{k}$. With one million sites, $k=5$ and $\varepsilon=1/2$, the exchange grid left 252 subsets on clustered inputs and about $9.3\times10^{26}$ on uniform inputs. The method extends to metrics of bounded doubling dimension and to compact domains with effective geometric routines, including filled simple polygons. At $s=2$, exact decision on metrics with nonzero distances in $\{1,2\}$ is W[1]-complete in $k$. A gap that shrinks with $k$ rules out a deterministic EPAS on unrestricted metrics unless FPT = W[1].

Authors: Michael Emmerich, Soumen Atta

We choose $k$ of $n$ sites to minimize Riesz $s$-energy, in which a pair at distance $t$ contributes $t^{-s}$. For $s=2$, exact selection is polynomial on a line but NP-hard in the plane. Adapting the proxy lemma of Ceccarello et al. gives an efficient parameterized approximation scheme (EPAS) for fixed Euclidean dimension and fixed $s>0$. To sharpen the bound, we start with a geometric packing. Farthest-first traversal gives $2k$ centers separated by at least the last insertion distance $r$. If one site of an optimum is removed, each of the $k-1$ remaining sites excludes at most one center closer than $r/2$, leaving at least $k+1$ centers free. The new step is a discrete exchange bound: the removed site's interaction energy is at most the average it would contribute at those free centers. In the plane at $s=2$, this reduces the representative bound from $O(k^2\log(2k)/\varepsilon^2)$ for a packing estimate of total energy to $O(k\log(2k)/\varepsilon^2)$, independently of $n$. Rational planar examples show that the resulting separation scale $r/\sqrt{\log(2k)}$ is sharp. A $(1+\varepsilon)$ approximation on rational planar sites takes $O(nk+n\log n)+(C\log(2k)/\varepsilon^2)^k$ operations of polynomial bit cost. The observed number $M$ of representatives determines the search size $\binom{M}{k}$. With one million sites, $k=5$ and $\varepsilon=1/2$, the exchange grid left 252 subsets on clustered inputs and about $9.3\times10^{26}$ on uniform inputs. The method extends to metrics of bounded doubling dimension and to compact domains with effective geometric routines, including filled simple polygons. At $s=2$, exact decision on metrics with nonzero distances in $\{1,2\}$ is W[1]-complete in $k$. A gap that shrinks with $k$ rules out a deterministic EPAS on unrestricted metrics unless FPT = W[1].

Wednesday, November 11 2026, 01:00

Pair-Merge Tokenisation Is Hard to Approximate on Binary Strings and Strongly NP-Hard on Unary Datasets

Authors: Kundan Baliga, Aiden Xie, Aditya Byju

Tokenisers such as Byte-Pair Encoding (BPE) shorten text by repeatedly merging an adjacent pair of symbols into a new one; fewer tokens mean less computation. We ask whether an efficient algorithm can choose $k$ merges that leave a string of length $n$ almost as short as possible; doing nothing is already within a factor $n$. We prove that, unless P = NP, no polynomial-time algorithm comes within a factor $c\,n/\log n$, for some constant $c>0$, already for one binary string and for BPE-style merges that replace every occurrence. The key idea is that $k$ merges form a grammar with $k$ rules: a vertex-cover gap becomes a grammar-size gap, which we amplify with relabelled copies, encode in binary and realise with explicit merges. For merges that may skip occurrences, a simple algorithm achieves $O(n/\log\log n)$, and no Karp reduction can prove hardness beyond order $n/\log n$ unless coNP $\subseteq$ NP/poly. Over a one-letter alphabet, both tokenisation problems left open by Kastreva et al. are strongly NP-complete on datasets with repeated strings.

Authors: Kundan Baliga, Aiden Xie, Aditya Byju

Tokenisers such as Byte-Pair Encoding (BPE) shorten text by repeatedly merging an adjacent pair of symbols into a new one; fewer tokens mean less computation. We ask whether an efficient algorithm can choose $k$ merges that leave a string of length $n$ almost as short as possible; doing nothing is already within a factor $n$. We prove that, unless P = NP, no polynomial-time algorithm comes within a factor $c\,n/\log n$, for some constant $c>0$, already for one binary string and for BPE-style merges that replace every occurrence. The key idea is that $k$ merges form a grammar with $k$ rules: a vertex-cover gap becomes a grammar-size gap, which we amplify with relabelled copies, encode in binary and realise with explicit merges. For merges that may skip occurrences, a simple algorithm achieves $O(n/\log\log n)$, and no Karp reduction can prove hardness beyond order $n/\log n$ unless coNP $\subseteq$ NP/poly. Over a one-letter alphabet, both tokenisation problems left open by Kastreva et al. are strongly NP-complete on datasets with repeated strings.

Wednesday, November 11 2026, 01:00

A Fast Algorithm for Subspace Convolution

Authors: Shlomo A. Perles, Omri Weinstein

A cornerstone result in Boolean lattice theory is the subset-zeta transform, which computes the $f$-weighted subset summation problem $(Z f)(B)=\sum_{A\subseteq B}f(A)$ for all $ B\subseteq[n]$ in $O(N \log N)$ additions instead of the naïve $O(3^n) = O(N^{\log_2 3})$, where $N=2^n$. This paper studies the algebraic analogue of this problem--the subspace summation problem--where subsets are replaced by linear subspaces in $\mathbb{F}_q^n$. Surprisingly, we show that this problem admits a similar $Θ_q(N_q \log N_q)$-time algorithm, where $N_q$ is the number of subspaces. This improves substantially (in both theory and practice) over lattice-based algorithms ([BHK+16, KKW16]), which are known to be optimal for addition-only algorithms over $\mathbb R$. We prove similar runtimes for inversion, rank-constrained sums and span convolution. We describe two main applications: Fast linear algebra on Grassmann graphs: Matrix--vector multiplication for arbitrary distance-invariant kernels on $N$-vertex Grassmann graphs in $O_q(N\log N)$ arithmetic operations, together with fast spectral projections and exact solutions of the corresponding linear systems. Graph coloring of low binary cut-rank graphs: An $n^{2^{r^2/4+O(1)}}$-time algorithm for computing the chromatic number and chromatic polynomial of an $n$-vertex graph, given a rank-decomposition of width $r$. We also give quantitative reductions from satisfiability and monotone constraint problems. Experiments show speedups in measured running time of up to $18$ over the covering-pair algorithm for the zeta transform, $190$ over Poisson summation for rank-constrained sums, and $22$ over the distance-matrix recurrence for Grassmann kernel multiplication. Lean 4 formalizations of the transform identities, arithmetic operation bounds, and graph-coloring constructions accompany the paper [PW26].

Authors: Shlomo A. Perles, Omri Weinstein

A cornerstone result in Boolean lattice theory is the subset-zeta transform, which computes the $f$-weighted subset summation problem $(Z f)(B)=\sum_{A\subseteq B}f(A)$ for all $ B\subseteq[n]$ in $O(N \log N)$ additions instead of the naïve $O(3^n) = O(N^{\log_2 3})$, where $N=2^n$. This paper studies the algebraic analogue of this problem--the subspace summation problem--where subsets are replaced by linear subspaces in $\mathbb{F}_q^n$. Surprisingly, we show that this problem admits a similar $Θ_q(N_q \log N_q)$-time algorithm, where $N_q$ is the number of subspaces. This improves substantially (in both theory and practice) over lattice-based algorithms ([BHK+16, KKW16]), which are known to be optimal for addition-only algorithms over $\mathbb R$. We prove similar runtimes for inversion, rank-constrained sums and span convolution. We describe two main applications: Fast linear algebra on Grassmann graphs: Matrix--vector multiplication for arbitrary distance-invariant kernels on $N$-vertex Grassmann graphs in $O_q(N\log N)$ arithmetic operations, together with fast spectral projections and exact solutions of the corresponding linear systems. Graph coloring of low binary cut-rank graphs: An $n^{2^{r^2/4+O(1)}}$-time algorithm for computing the chromatic number and chromatic polynomial of an $n$-vertex graph, given a rank-decomposition of width $r$. We also give quantitative reductions from satisfiability and monotone constraint problems. Experiments show speedups in measured running time of up to $18$ over the covering-pair algorithm for the zeta transform, $190$ over Poisson summation for rank-constrained sums, and $22$ over the distance-matrix recurrence for Grassmann kernel multiplication. Lean 4 formalizations of the transform identities, arithmetic operation bounds, and graph-coloring constructions accompany the paper [PW26].

Wednesday, November 11 2026, 01:00

Online Graph Coloring: Competitive Ratios Separating Randomized and Deterministic Algorithms

Authors: Ken-ichi Kawarabayashi, Hirotaka Yoneda, Masataka Yoneda

We study online coloring, focusing on the competitive ratio. The best previously known results were (1) a lower bound of $Ω(n / \log^2 n)$ for both deterministic and randomized algorithms by Halldórsson and Szegedy (1994), (2) an upper bound of $O(n / \log n)$ for randomized algorithms by Halldórsson (1997), and (3) an upper bound of $O(n / \log \log n)$ for deterministic algorithms by Kawarabayashi, Yoneda, and Yoneda (2025). In this paper, we improve these results as follows: 1. A new lower bound of $Ω(n / \log n)$ for deterministic algorithms. 2. A new upper bound of $O(n (\log \log n)^2 / \log^2 n)$ for randomized algorithms. 3. A new upper bound of $O(n / (\log \log n)^{2-\varepsilon})$ for deterministic algorithms for every fixed $\varepsilon>0$. For randomized algorithms, we nearly close the gap between the lower and upper bounds; the remaining gap is only $O((\log \log n)^2)$. We establish the asymptotic separation between randomized and deterministic online graph coloring. Against an oblivious adversary, a polynomial-time randomized algorithm achieves a competitive ratio asymptotically smaller than any deterministic algorithm can guarantee, even with unlimited computation.

Authors: Ken-ichi Kawarabayashi, Hirotaka Yoneda, Masataka Yoneda

We study online coloring, focusing on the competitive ratio. The best previously known results were (1) a lower bound of $Ω(n / \log^2 n)$ for both deterministic and randomized algorithms by Halldórsson and Szegedy (1994), (2) an upper bound of $O(n / \log n)$ for randomized algorithms by Halldórsson (1997), and (3) an upper bound of $O(n / \log \log n)$ for deterministic algorithms by Kawarabayashi, Yoneda, and Yoneda (2025). In this paper, we improve these results as follows: 1. A new lower bound of $Ω(n / \log n)$ for deterministic algorithms. 2. A new upper bound of $O(n (\log \log n)^2 / \log^2 n)$ for randomized algorithms. 3. A new upper bound of $O(n / (\log \log n)^{2-\varepsilon})$ for deterministic algorithms for every fixed $\varepsilon>0$. For randomized algorithms, we nearly close the gap between the lower and upper bounds; the remaining gap is only $O((\log \log n)^2)$. We establish the asymptotic separation between randomized and deterministic online graph coloring. Against an oblivious adversary, a polynomial-time randomized algorithm achieves a competitive ratio asymptotically smaller than any deterministic algorithm can guarantee, even with unlimited computation.

Wednesday, November 11 2026, 01:00

A note on the intractability of enumerating the maximal satisfiable subformulas of a Horn expression

Authors: Oscar Defrain, Corentin Jazeron, Kazuhisa Makino

We show that, given a Horn expression and a family of inclusion-wise maximal satisfiable subsets of its clauses, deciding whether there exists another maximal satisfiable subset is NP-complete. As a consequence, unless P=NP, there is no output-polynomial time algorithm enumerating the maximal satisfiable subformulas of a Horn expression.

Authors: Oscar Defrain, Corentin Jazeron, Kazuhisa Makino

We show that, given a Horn expression and a family of inclusion-wise maximal satisfiable subsets of its clauses, deciding whether there exists another maximal satisfiable subset is NP-complete. As a consequence, unless P=NP, there is no output-polynomial time algorithm enumerating the maximal satisfiable subformulas of a Horn expression.

Wednesday, November 11 2026, 01:00

Directed Singular-Value Sparsifiers with $O(n \log n/ε^2)$ Arcs

Authors: Emile Anand

Given an unweighted directed graph, a sparsifier is a nonnegative reweighting of original arcs that preserves every in-degree and out-degree exactly. In this work, we study singular-value sparsifiers, which are generalizations of spectral sparsifiers. Singular-value sparsifiers find a number of applications in theoretical computer science, such as maintaining directed cuts and flows, speeding up directed Laplacian solvers, and sparsifying dense weight matrices in neural networks. Strong singular-value (SV) approximation supports sparse representations of directed random walks with guarantees that remain stable under products, and controls bilinear error in the left and right Schur-complement metrics. We prove that every unweighted directed graph on $n$ vertices admits an $\varepsilon$-SV sparsifier with $O(n\log(2n)/\varepsilon^2)$ arcs, improving the $O(n\log^5 n/\varepsilon^2)$ support bound of Lau, Wang, and Zhou. The sparsifier is a nonnegative reweighting of original arcs and preserves every in-degree and out-degree exactly. We also give a randomized construction with $\tilde{O}(m+n)$ real-arithmetic running time, where $m$ is the input support size, and the output has the same support bound and rational weights of polynomial bit length.

Authors: Emile Anand

Given an unweighted directed graph, a sparsifier is a nonnegative reweighting of original arcs that preserves every in-degree and out-degree exactly. In this work, we study singular-value sparsifiers, which are generalizations of spectral sparsifiers. Singular-value sparsifiers find a number of applications in theoretical computer science, such as maintaining directed cuts and flows, speeding up directed Laplacian solvers, and sparsifying dense weight matrices in neural networks. Strong singular-value (SV) approximation supports sparse representations of directed random walks with guarantees that remain stable under products, and controls bilinear error in the left and right Schur-complement metrics. We prove that every unweighted directed graph on $n$ vertices admits an $\varepsilon$-SV sparsifier with $O(n\log(2n)/\varepsilon^2)$ arcs, improving the $O(n\log^5 n/\varepsilon^2)$ support bound of Lau, Wang, and Zhou. The sparsifier is a nonnegative reweighting of original arcs and preserves every in-degree and out-degree exactly. We also give a randomized construction with $\tilde{O}(m+n)$ real-arithmetic running time, where $m$ is the input support size, and the output has the same support bound and rational weights of polynomial bit length.

Wednesday, November 11 2026, 01:00

Bounding and Approximating Kostka Numbers

Authors: Jonathan Leake, Zirou Zhang

Kostka numbers, the coefficients of Schur polynomials, count semistandard Young tableaux of prescribed shape and weight. We give a deterministic algorithm for approximating Kostka numbers within a simply exponential factor in the number of variables and the size of the partition. Our approach combines Gurvits capacity bounds for log-concave polynomials with an algorithm for approximating the Gurvits capacity of a Schur polynomial. For a Schur polynomial in $n$ variables indexed by $λ$, we show that the logarithm of the capacity associated with any nonzero coefficient can be approximated to additive error $ε$ in time polynomial in $n$, $|λ|$, and $\log(1/ε)$. The analysis establishes an explicit bounded optimization region and efficient approximate first-order oracles, and handles boundary weights through a factorization into smaller Schur polynomials.

Authors: Jonathan Leake, Zirou Zhang

Kostka numbers, the coefficients of Schur polynomials, count semistandard Young tableaux of prescribed shape and weight. We give a deterministic algorithm for approximating Kostka numbers within a simply exponential factor in the number of variables and the size of the partition. Our approach combines Gurvits capacity bounds for log-concave polynomials with an algorithm for approximating the Gurvits capacity of a Schur polynomial. For a Schur polynomial in $n$ variables indexed by $λ$, we show that the logarithm of the capacity associated with any nonzero coefficient can be approximated to additive error $ε$ in time polynomial in $n$, $|λ|$, and $\log(1/ε)$. The analysis establishes an explicit bounded optimization region and efficient approximate first-order oracles, and handles boundary weights through a factorization into smaller Schur polynomials.

Wednesday, November 11 2026, 01:00

Nonadaptive One-Sided Testing of Binary Rank

Authors: Nader H. Bshouty

The binary rank of a binary matrix is the minimum number of disjoint all-one rectangles that partition its one-entries. We study entry-query testing of binary rank at most $d$, with one-sided error and nonadaptive queries. We give a tester using $\tilde O(d^4/ε^2)$ queries, independent of the matrix dimensions and a general lower bound of $\tilde Ω(d^{3/2}/ε)$.

Authors: Nader H. Bshouty

The binary rank of a binary matrix is the minimum number of disjoint all-one rectangles that partition its one-entries. We study entry-query testing of binary rank at most $d$, with one-sided error and nonadaptive queries. We give a tester using $\tilde O(d^4/ε^2)$ queries, independent of the matrix dimensions and a general lower bound of $\tilde Ω(d^{3/2}/ε)$.

Wednesday, November 11 2026, 01:00

Mixed-integer dual network problems are solvable in strongly polynomial time

Authors: Giacomo Zambelli

In this paper we show that mixed-integer linear programming is solvable in strongly polynomial time if the constraint matrix is totally unimodular with at most two nonzero entries in every row. This is done by reducing the problem to an instance of minimization of an $L$-convex function.

Authors: Giacomo Zambelli

In this paper we show that mixed-integer linear programming is solvable in strongly polynomial time if the constraint matrix is totally unimodular with at most two nonzero entries in every row. This is done by reducing the problem to an instance of minimization of an $L$-convex function.

Wednesday, November 11 2026, 01:00

Bounded Weighted Edit Distance with Metric Costs

Authors: Jihan Wang

We give a deterministic $O(n+k^2)$-time algorithm for bounded weighted edit distance when the input costs form an integer metric on the alphabet extended by a gap symbol. Here $n$ is the total input length and $k$ is the given threshold on the distance. If the distance is at most $k$, the algorithm returns it with an optimal alignment. Otherwise, it reports that the distance exceeds $k$. The running time matches the classical unit-cost bound and is independent of the alphabet size and of the largest cost. The algorithm keeps only the farthest-reaching prefix alignment for each cost and each difference between the total gap costs of the aligned prefixes. For real metrics whose nonzero costs are at least one, scaling and upward rounding give a $(1+\varepsilon)$-approximation for the bounded problem in $\widetilde{O}(n+(k/\varepsilon)^2)$ time. If insertions and deletions have integer costs, the running time improves to $\widetilde{O}(n+k^2/\varepsilon)$.

Authors: Jihan Wang

We give a deterministic $O(n+k^2)$-time algorithm for bounded weighted edit distance when the input costs form an integer metric on the alphabet extended by a gap symbol. Here $n$ is the total input length and $k$ is the given threshold on the distance. If the distance is at most $k$, the algorithm returns it with an optimal alignment. Otherwise, it reports that the distance exceeds $k$. The running time matches the classical unit-cost bound and is independent of the alphabet size and of the largest cost. The algorithm keeps only the farthest-reaching prefix alignment for each cost and each difference between the total gap costs of the aligned prefixes. For real metrics whose nonzero costs are at least one, scaling and upward rounding give a $(1+\varepsilon)$-approximation for the bounded problem in $\widetilde{O}(n+(k/\varepsilon)^2)$ time. If insertions and deletions have integer costs, the running time improves to $\widetilde{O}(n+k^2/\varepsilon)$.

Wednesday, November 11 2026, 01:00

What to Predict for Efficient Scheduling on Parallel Machines

Authors: Evripidis Bampis, Bruno Escoffier, Dimitris Fotakis, Giorgos Mitropoulos, Michalis Xefteris

We study learning-augmented algorithms for scheduling on multiple parallel machines under the objectives of minimizing the makespan or the total weighted completion time. We assume that the algorithm has access to noisy information (the prediction) about an optimal solution, expressed through a solution encoding that implicitly quantifies both the amount of information provided and the nature of noise (errors in the prediction). Our central question is to determine the maximum level of noise that still allows for efficiently recovering this optimal solution. We observe that information guiding greedy scheduling to optimality or revealing the optimal assignment of tasks to machines is very sensitive to noise, and prove that recovering an optimal solution from solution encodings capturing either is computationally hard, even for a small (non-constant) number of errors. On the positive side, we show that an order-based encoding, which leads to errors that are dispersed locally, is robust to noise. We present a dynamic programming framework that utilizes this encoding and solves optimally several classical scheduling problems, including makespan and total weighted completion time minimization on identical or unrelated machines.

Authors: Evripidis Bampis, Bruno Escoffier, Dimitris Fotakis, Giorgos Mitropoulos, Michalis Xefteris

We study learning-augmented algorithms for scheduling on multiple parallel machines under the objectives of minimizing the makespan or the total weighted completion time. We assume that the algorithm has access to noisy information (the prediction) about an optimal solution, expressed through a solution encoding that implicitly quantifies both the amount of information provided and the nature of noise (errors in the prediction). Our central question is to determine the maximum level of noise that still allows for efficiently recovering this optimal solution. We observe that information guiding greedy scheduling to optimality or revealing the optimal assignment of tasks to machines is very sensitive to noise, and prove that recovering an optimal solution from solution encodings capturing either is computationally hard, even for a small (non-constant) number of errors. On the positive side, we show that an order-based encoding, which leads to errors that are dispersed locally, is robust to noise. We present a dynamic programming framework that utilizes this encoding and solves optimally several classical scheduling problems, including makespan and total weighted completion time minimization on identical or unrelated machines.

Wednesday, November 11 2026, 01:00

Real-Time String Indexing in Optimal Time

Authors: Gregory Kucherov, Yakov Nekrich

In this paper we describe the first fully real-time indexing solution for pattern matching over an arbitrary integer alphabet. Our data structure spends $O(1)$ worst-case time w.h.p. for processing each new letter appended to $T$ and worst-case $O(|P|+occ)$ time for reporting all occurrences of $P$, where $T$ and $P$ are strings over an integer alphabet. The main idea of our solution is a novel representation of the suffix tree, the path-decomposed suffix tree, which does not store the full topology of the suffix tree but instead decomposes it into paths, each stored in a dictionary that encodes parent--child relationships indirectly.

Authors: Gregory Kucherov, Yakov Nekrich

In this paper we describe the first fully real-time indexing solution for pattern matching over an arbitrary integer alphabet. Our data structure spends $O(1)$ worst-case time w.h.p. for processing each new letter appended to $T$ and worst-case $O(|P|+occ)$ time for reporting all occurrences of $P$, where $T$ and $P$ are strings over an integer alphabet. The main idea of our solution is a novel representation of the suffix tree, the path-decomposed suffix tree, which does not store the full topology of the suffix tree but instead decomposes it into paths, each stored in a dictionary that encodes parent--child relationships indirectly.

Wednesday, November 11 2026, 01:00

Online Network Design with Maximum Delay and Stochastic Arrivals

Authors: Tianhang Lu

We study online network services that combine pending requests into batches. Each batch pays for its spatial service and for the waiting time of its oldest request. The objective is the sum of these costs over all batches. Thus requests can share both a spatial purchase and one maximum-delay charge. We assume independent Poisson arrivals of known rates, while actual requests are revealed only upon arrival. For stateless services, suppose that each type has an attainable minimum cost when served alone, that every batch costs at least the stand-alone minimum of each type it contains, and that one finite-cost action can serve any batch, regardless of its request multiplicities. We give a deterministic algorithm with long-run ratio of expectations $e/(e-1)$ against an unrestricted offline optimum. This factor is optimal, even allowing randomization, already at one request location. The key comparison measures the union of time intervals associated with requests. The theorem applies directly to multi-root and pair aggregation on trees. We also study facility location, where each request pays its own connection distance and the bounded-action assumption can fail. A deterministic policy gives a ratio $3$ for arbitrary nonnegative opening costs. It immediately serves a finite initial set of requests, computes an assignment from their observed arrival gaps, and then keeps that assignment fixed. Its analysis combines Poisson insertion with a time-slice packing bound on the offline optimum. Every guarantee includes the full cost of completing finite inputs without an end-of-input signal.

Authors: Tianhang Lu

We study online network services that combine pending requests into batches. Each batch pays for its spatial service and for the waiting time of its oldest request. The objective is the sum of these costs over all batches. Thus requests can share both a spatial purchase and one maximum-delay charge. We assume independent Poisson arrivals of known rates, while actual requests are revealed only upon arrival. For stateless services, suppose that each type has an attainable minimum cost when served alone, that every batch costs at least the stand-alone minimum of each type it contains, and that one finite-cost action can serve any batch, regardless of its request multiplicities. We give a deterministic algorithm with long-run ratio of expectations $e/(e-1)$ against an unrestricted offline optimum. This factor is optimal, even allowing randomization, already at one request location. The key comparison measures the union of time intervals associated with requests. The theorem applies directly to multi-root and pair aggregation on trees. We also study facility location, where each request pays its own connection distance and the bounded-action assumption can fail. A deterministic policy gives a ratio $3$ for arbitrary nonnegative opening costs. It immediately serves a finite initial set of requests, computes an assignment from their observed arrival gaps, and then keeps that assignment fixed. Its analysis combines Poisson insertion with a time-slice packing bound on the offline optimum. Every guarantee includes the full cost of completing finite inputs without an end-of-input signal.

Wednesday, November 11 2026, 01:00

Online Network Design with Additive Delays and Poisson Arrivals

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

We study online service systems in which requests arrive at locations in a network. A service may open a facility and connect customers, buy a subtree linking them to a root, or send a mobile server to visit them. Combining requests can save opening, network, or travel costs, but each request pays its own waiting time. The objective is the sum of these spatial costs and individual delays. We assume independent Poisson arrivals of known rates. The algorithms know how often requests occur, but observe them only upon arrival. We show that these rates suffice to make fixed spatial choices that compete with an offline optimum free to adapt to the realized input. We obtain long-run ratios of expectations below $27.249$ for facility location in finite metrics, $94.415$ for multi-root aggregation on trees, and $24.326$ for online service on trees. All three algorithms are deterministic; the multi-root algorithm also has polynomial-time preprocessing on rational inputs. The proofs connect rate-based spatial choices with lower bounds on unavoidable purchase, movement, and waiting costs. All main proofs are self-contained, and every guarantee includes the full cost of completing finite inputs.

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

We study online service systems in which requests arrive at locations in a network. A service may open a facility and connect customers, buy a subtree linking them to a root, or send a mobile server to visit them. Combining requests can save opening, network, or travel costs, but each request pays its own waiting time. The objective is the sum of these spatial costs and individual delays. We assume independent Poisson arrivals of known rates. The algorithms know how often requests occur, but observe them only upon arrival. We show that these rates suffice to make fixed spatial choices that compete with an offline optimum free to adapt to the realized input. We obtain long-run ratios of expectations below $27.249$ for facility location in finite metrics, $94.415$ for multi-root aggregation on trees, and $24.326$ for online service on trees. All three algorithms are deterministic; the multi-root algorithm also has polynomial-time preprocessing on rational inputs. The proofs connect rate-based spatial choices with lower bounds on unavoidable purchase, movement, and waiting costs. All main proofs are self-contained, and every guarantee includes the full cost of completing finite inputs.

Wednesday, November 11 2026, 01:00

Stochastic Aggregation with Prescribed Resources and Additive Delay

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

We study online aggregation with prescribed resources and additive delay. In this problem, we are given a finite set of resources with purchase costs and a collection of request types, each specifying the resources it requires. Requests arrive over time, and each request can be served immediately or postponed to a later time. A service may serve several pending requests together by purchasing every resource required by at least one of them, paying for each purchased resource only once. These purchases serve only the current batch; resources needed in a later service must be purchased again. A request served $t$ units of time after its arrival incurs delay cost $t$. The objective is to serve all requests while minimizing the sum of resource purchase costs and individual delay costs. We consider a stochastic version of this problem in which requests of each type follow an independent Poisson process of known rate. An online algorithm knows these rates but observes requests only upon arrival. We compare its expected cost with that of an offline optimum that knows the entire realized input and may choose every batch freely. A resource--scale linear program supplies both an offline lower certificate and synchronized service periods. This yields a deterministic online algorithm with polynomial-time preprocessing on rational input and a long-run ratio of expectations below $2.113$. The guarantee applies in particular to rooted multi-level aggregation and prescribed-pair aggregation on trees. Besides, for Poisson TCP acknowledgment, we further characterize the optimal long-run ratio of expectations at every load, showing that its worst-case value is approximately 1.409 and is attained by a deterministic online policy.

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

We study online aggregation with prescribed resources and additive delay. In this problem, we are given a finite set of resources with purchase costs and a collection of request types, each specifying the resources it requires. Requests arrive over time, and each request can be served immediately or postponed to a later time. A service may serve several pending requests together by purchasing every resource required by at least one of them, paying for each purchased resource only once. These purchases serve only the current batch; resources needed in a later service must be purchased again. A request served $t$ units of time after its arrival incurs delay cost $t$. The objective is to serve all requests while minimizing the sum of resource purchase costs and individual delay costs. We consider a stochastic version of this problem in which requests of each type follow an independent Poisson process of known rate. An online algorithm knows these rates but observes requests only upon arrival. We compare its expected cost with that of an offline optimum that knows the entire realized input and may choose every batch freely. A resource--scale linear program supplies both an offline lower certificate and synchronized service periods. This yields a deterministic online algorithm with polynomial-time preprocessing on rational input and a long-run ratio of expectations below $2.113$. The guarantee applies in particular to rooted multi-level aggregation and prescribed-pair aggregation on trees. Besides, for Poisson TCP acknowledgment, we further characterize the optimal long-run ratio of expectations at every load, showing that its worst-case value is approximately 1.409 and is attained by a deterministic online policy.

Wednesday, November 11 2026, 01:00

List Update with Per-Page Maximum Delay

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

We study list update with per-page maximum delay. Requests may wait, and one access clears all pending requests of a page at its current list rank, paying their maximum waiting penalty. Each service can change the permutation and therefore the cost of later accesses. We give a strictly $5$-competitive deterministic algorithm: serve a page when the oldest waiting time reaches two thirds of its current rank, then move it to the front. Our main tool is an asynchronous marked-rank coupling that compares different list orders and different service histories. A temporal transfer principle separates this permutation argument from two waiting-cost inequalities. It gives factor $5$ for page-specific superadditive penalties, common calendar rates, and additional hard deadlines; deadlines alone admit factor $3$. Two structural conditions on request-specific penalties give factor $3+2κ$. In particular, nonincreasing request severity retains factor $5$ even with unbounded weight ratios. For linear delay, offline optimization is NP-complete but fixed-parameter tractable in the number of pages. We characterize fixed-list batching, show an $Ω(m)$ gap between the best static order and dynamic reordering, and prove that the optimal deterministic factor for one page is exactly $2$. Counterexamples delineate the threshold rule's scope.

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

We study list update with per-page maximum delay. Requests may wait, and one access clears all pending requests of a page at its current list rank, paying their maximum waiting penalty. Each service can change the permutation and therefore the cost of later accesses. We give a strictly $5$-competitive deterministic algorithm: serve a page when the oldest waiting time reaches two thirds of its current rank, then move it to the front. Our main tool is an asynchronous marked-rank coupling that compares different list orders and different service histories. A temporal transfer principle separates this permutation argument from two waiting-cost inequalities. It gives factor $5$ for page-specific superadditive penalties, common calendar rates, and additional hard deadlines; deadlines alone admit factor $3$. Two structural conditions on request-specific penalties give factor $3+2κ$. In particular, nonincreasing request severity retains factor $5$ even with unbounded weight ratios. For linear delay, offline optimization is NP-complete but fixed-parameter tractable in the number of pages. We characterize fixed-list batching, show an $Ω(m)$ gap between the best static order and dynamic reordering, and prove that the optimal deterministic factor for one page is exactly $2$. Counterexamples delineate the threshold rule's scope.

Wednesday, November 11 2026, 01:00

Approximating graphic TSP by b-matchings

Authors: Swati Gupta, Mehrdad Sohrabi, Levent Tunçel

We give a polynomial-time algorithm for graphic TSP that returns a tour of length at most $\frac{89}{64}=\frac{7}{5}-\frac{3}{320}$ times the value of the subtour relaxation. This improves the long-standing ratio $\frac75$ of Sebő and Vygen and shows that the integrality gap of the subtour LP on graphic instances is at most $\frac{89}{64}\approx 1.3906$. We optimize a non-linear objective function over an odd-join polytope, which uses (i) a new removable system of edges, and (ii) reconnection of subtours if the graph is disconnected. This approach gives a $\frac{13}{10}$-approximation algorithm for $4$-regular $4$-vertex-connected graphs, improving the $\frac43$ of Newman, and a $\frac43$-approximation for $k$-vertex-connected graphs of maximum degree at most $k+1$, for every $k\ge 3$. For the general graphic TSP, we construct a feasible point in the odd-join polytope, using an optimal vertex from the Subtour linear programming relaxation of TSP, which helps us bound the approximation factor by 89/64.

Authors: Swati Gupta, Mehrdad Sohrabi, Levent Tunçel

We give a polynomial-time algorithm for graphic TSP that returns a tour of length at most $\frac{89}{64}=\frac{7}{5}-\frac{3}{320}$ times the value of the subtour relaxation. This improves the long-standing ratio $\frac75$ of Sebő and Vygen and shows that the integrality gap of the subtour LP on graphic instances is at most $\frac{89}{64}\approx 1.3906$. We optimize a non-linear objective function over an odd-join polytope, which uses (i) a new removable system of edges, and (ii) reconnection of subtours if the graph is disconnected. This approach gives a $\frac{13}{10}$-approximation algorithm for $4$-regular $4$-vertex-connected graphs, improving the $\frac43$ of Newman, and a $\frac43$-approximation for $k$-vertex-connected graphs of maximum degree at most $k+1$, for every $k\ge 3$. For the general graphic TSP, we construct a feasible point in the odd-join polytope, using an optimal vertex from the Subtour linear programming relaxation of TSP, which helps us bound the approximation factor by 89/64.

Wednesday, November 11 2026, 01:00

Greedy+Singleton for knapsack-constrained monotone submodular maximization has approximation ratio exactly $β^\star \approx 0.42944$

Authors: Kundan Baliga, Aiden Xie, Aditya Byju

Greedy+Singleton is the textbook heuristic for monotone submodular maximization under a knapsack constraint: it returns the better of the cost-benefit greedy set and the best single item. Its worst-case approximation ratio, known only to lie in $[0.427, 0.42945]$ (Feldman-Nutov-Shoham; Kulik-Schwartz-Shachnai), is exactly an explicit constant $β^\star\approx 0.42944$. In the extremal instances, many cheap "filler" items wear down the value of two large optimal items while greedy spends the budget, until neither fits. This picture gives a sharper factor-revealing relaxation; we bound it analytically near the extremal parameters, up to finitely many interval-arithmetic checks, and certify it elsewhere by an exact branch-and-bound. The matching bad instances are tie-free and checkable by hand. For the version that stops at the first item that does not fit, Wolsey's bound $1-e^{-β_W}\approx 0.35780$, where $e^{β_W}=2-β_W$, is tight; Wolsey stated this for his variant without giving the instances, and we give an explicit family with no ties that also covers comparison with the best feasible singleton. The CELF rule also has ratio exactly $β^\star$.

Authors: Kundan Baliga, Aiden Xie, Aditya Byju

Greedy+Singleton is the textbook heuristic for monotone submodular maximization under a knapsack constraint: it returns the better of the cost-benefit greedy set and the best single item. Its worst-case approximation ratio, known only to lie in $[0.427, 0.42945]$ (Feldman-Nutov-Shoham; Kulik-Schwartz-Shachnai), is exactly an explicit constant $β^\star\approx 0.42944$. In the extremal instances, many cheap "filler" items wear down the value of two large optimal items while greedy spends the budget, until neither fits. This picture gives a sharper factor-revealing relaxation; we bound it analytically near the extremal parameters, up to finitely many interval-arithmetic checks, and certify it elsewhere by an exact branch-and-bound. The matching bad instances are tie-free and checkable by hand. For the version that stops at the first item that does not fit, Wolsey's bound $1-e^{-β_W}\approx 0.35780$, where $e^{β_W}=2-β_W$, is tight; Wolsey stated this for his variant without giving the instances, and we give an explicit family with no ties that also covers comparison with the best feasible singleton. The CELF rule also has ratio exactly $β^\star$.

Wednesday, November 11 2026, 01:00

Improved Bounds for Online Matching with Random Edge Arrivals

Authors: Yilong Feng, Haolong Li, Zonghan Yang

We study the unweighted online matching problem with random edge arrivals, where each edge arrives at an independent time drawn uniformly from $[0,1]$, and the algorithm must immediately and irrevocably accept or reject it while maintaining a matching. Guruganesh and Singla (IPCO, 2017) first broke the $1/2$ barrier of the naive Greedy algorithm by giving a $(1/2+\varepsilon)$-competitive algorithm for a small constant $\varepsilon>0$, a guarantee that has remained the best known for nearly a decade. We give a simple three-phase algorithm that achieves a competitive ratio of $0.5445$ by deliberately rejecting selected early edges to reserve their endpoints for later matching opportunities. With a different choice of parameters, the same algorithm is $0.5535$-competitive on triangle-free graphs, including bipartite graphs. We also establish competitive ratio upper bounds below $0.6964$ for general graphs and $0.7043$ for bipartite graphs via exact rational evaluation of an optimal-policy recurrence on fixed graphs, even when the input graph is known up to isomorphism.

Authors: Yilong Feng, Haolong Li, Zonghan Yang

We study the unweighted online matching problem with random edge arrivals, where each edge arrives at an independent time drawn uniformly from $[0,1]$, and the algorithm must immediately and irrevocably accept or reject it while maintaining a matching. Guruganesh and Singla (IPCO, 2017) first broke the $1/2$ barrier of the naive Greedy algorithm by giving a $(1/2+\varepsilon)$-competitive algorithm for a small constant $\varepsilon>0$, a guarantee that has remained the best known for nearly a decade. We give a simple three-phase algorithm that achieves a competitive ratio of $0.5445$ by deliberately rejecting selected early edges to reserve their endpoints for later matching opportunities. With a different choice of parameters, the same algorithm is $0.5535$-competitive on triangle-free graphs, including bipartite graphs. We also establish competitive ratio upper bounds below $0.6964$ for general graphs and $0.7043$ for bipartite graphs via exact rational evaluation of an optimal-policy recurrence on fixed graphs, even when the input graph is known up to isomorphism.

Wednesday, November 11 2026, 01:00

Minimum Membership Dominating Set and Its Generalization: Leafage vs. Asteroidal Number

Authors: Yuma Tamura, Daichi Wakayama, Akira Suzuki, Xiao Zhou

The Dominating Set problem is known to be fixed-parameter tractable when parameterized by leafage, a structural graph parameter defined on chordal graphs. We investigate whether this tractability extends to its variants with natural constraints. Specifically, we study the Minimum Membership Dominating Set problem and its generalization, the Min. $(σ, ρ)$-Dominating Set problem. Given a graph $G$ and a positive integer $\ell$, Minimum Membership Dominating Set asks whether there is a dominating set $D$ such that every vertex in $G$ is dominated by at most $\ell$ vertices in $D$. We first show a negative result: Minimum Membership Dominating Set parameterized by leafage plus $\ell$ is W[1]-hard, demonstrating that even this natural constraint serves as a severe barrier to designing FPT algorithms. To complement this, we present an XP algorithm for the Min. $(σ, ρ)$-Dominating Set problem parameterized by asteroidal number plus $μ$, where $σ$ and $ρ$ are finite sets of non-negative integers and $μ$ is the maximum value in $σ\cup ρ$. Note that asteroidal number generalizes leafage to arbitrary graphs. Combining this algorithm with our structural bound that every graph $G$ admits a dominating set such that no vertex is dominated by more than $7\mathsf{an}(G)-6$ vertices, where $\mathsf{an}(G)$ denotes the asteroidal number of $G$, we show that Minimum Membership Dominating Set admits an XP algorithm parameterized solely by asteroidal number.

Authors: Yuma Tamura, Daichi Wakayama, Akira Suzuki, Xiao Zhou

The Dominating Set problem is known to be fixed-parameter tractable when parameterized by leafage, a structural graph parameter defined on chordal graphs. We investigate whether this tractability extends to its variants with natural constraints. Specifically, we study the Minimum Membership Dominating Set problem and its generalization, the Min. $(σ, ρ)$-Dominating Set problem. Given a graph $G$ and a positive integer $\ell$, Minimum Membership Dominating Set asks whether there is a dominating set $D$ such that every vertex in $G$ is dominated by at most $\ell$ vertices in $D$. We first show a negative result: Minimum Membership Dominating Set parameterized by leafage plus $\ell$ is W[1]-hard, demonstrating that even this natural constraint serves as a severe barrier to designing FPT algorithms. To complement this, we present an XP algorithm for the Min. $(σ, ρ)$-Dominating Set problem parameterized by asteroidal number plus $μ$, where $σ$ and $ρ$ are finite sets of non-negative integers and $μ$ is the maximum value in $σ\cup ρ$. Note that asteroidal number generalizes leafage to arbitrary graphs. Combining this algorithm with our structural bound that every graph $G$ admits a dominating set such that no vertex is dominated by more than $7\mathsf{an}(G)-6$ vertices, where $\mathsf{an}(G)$ denotes the asteroidal number of $G$, we show that Minimum Membership Dominating Set admits an XP algorithm parameterized solely by asteroidal number.

Wednesday, November 11 2026, 01:00

Function Inversion Reporting

Authors: Itai Dinur, Alexander Golovnev, Chao Yan, Yiding Zhu

In the Function Inversion problem, the goal is to invert a function $f:[N] \to [N]$ in two phases. In the preprocessing phase, the offline algorithm computes an advice string of length $S$ bits. In the online phase, the online algorithm receives the advice and a challenge $y \in [N]$, and the goal is to find a preimage $x \in [N]$ such that $f(x) = y$ in time $T$. The problem was initially studied in cryptography, but in recent years it has found various applications to data structure problems. Yet, several of the applications require a variant of Function Inversion, known as Function Inversion Reporting, where the goal is to output all preimages of the challenge $y$. In this paper, we study the Function Inversion Reporting problem in a systematic way by unifying and improving existing algorithms. As a result, we obtain improved data structures for several problems including 3SUM-Indexing Reporting, Range Reporting, Approximate Nearest Neighbor, Jumbled Indexing Reporting, and Gapped String Indexing Reporting.

Authors: Itai Dinur, Alexander Golovnev, Chao Yan, Yiding Zhu

In the Function Inversion problem, the goal is to invert a function $f:[N] \to [N]$ in two phases. In the preprocessing phase, the offline algorithm computes an advice string of length $S$ bits. In the online phase, the online algorithm receives the advice and a challenge $y \in [N]$, and the goal is to find a preimage $x \in [N]$ such that $f(x) = y$ in time $T$. The problem was initially studied in cryptography, but in recent years it has found various applications to data structure problems. Yet, several of the applications require a variant of Function Inversion, known as Function Inversion Reporting, where the goal is to output all preimages of the challenge $y$. In this paper, we study the Function Inversion Reporting problem in a systematic way by unifying and improving existing algorithms. As a result, we obtain improved data structures for several problems including 3SUM-Indexing Reporting, Range Reporting, Approximate Nearest Neighbor, Jumbled Indexing Reporting, and Gapped String Indexing Reporting.

Wednesday, November 11 2026, 01:00

An Improved $(1+2e)$-Approximation Algorithm for Metric Data Placement

Authors: S. Rasoul Etesami

We study the metric data placement problem with site-capacity constraints and fixed installation costs, aiming to minimize the total cost of placing data (resources) and accessing their nearest copies. We develop a deterministic polynomial-time approximation algorithm based on a natural linear programming relaxation of the problem. Starting from an optimal fractional solution, we construct multiscale metric-ball partitions independently for each resource and connect overlapping balls at consecutive levels to form a flow network. The resource layers are then coupled through shared site-capacity arcs, allowing us to obtain an integral placement by minimizing a surrogate cost over a single min-cost circulation. By optimizing the choice of ball radii, we establish an approximation factor of $1+2e\approx 6.44$. To our knowledge, this is the best approximation guarantee for the metric data placement problem since its introduction by Baev and Rajaraman in 2001.

Authors: S. Rasoul Etesami

We study the metric data placement problem with site-capacity constraints and fixed installation costs, aiming to minimize the total cost of placing data (resources) and accessing their nearest copies. We develop a deterministic polynomial-time approximation algorithm based on a natural linear programming relaxation of the problem. Starting from an optimal fractional solution, we construct multiscale metric-ball partitions independently for each resource and connect overlapping balls at consecutive levels to form a flow network. The resource layers are then coupled through shared site-capacity arcs, allowing us to obtain an integral placement by minimizing a surrogate cost over a single min-cost circulation. By optimizing the choice of ball radii, we establish an approximation factor of $1+2e\approx 6.44$. To our knowledge, this is the best approximation guarantee for the metric data placement problem since its introduction by Baev and Rajaraman in 2001.

Wednesday, November 11 2026, 01:00

Beyond the First Hit: Towards Tight Guarantees for $k$-means++

Authors: Aravind Reddy

Let $R_k$ be the worst-case ratio between the expected cost of the $k$ seeds chosen by $k$-means++ and the optimal $k$-means cost. In their seminal paper introducing $k$-means++, Arthur and Vassilvitskii (SODA 2007) showed that $2\ln k\le R_k\le8(\ln k+2)$, and Makarychev, Reddy and Shan (NeurIPS 2020) improved the upper bound to $5(\ln k+2)$. We prove $R_k\le\frac{1163}{233}+2.108\,H_{k-2}$ for every $k\ge3$ by a computer-assisted argument, and $R_k\le5+2.3139\,H_{k-2}$ for every $k\ge2$ by a short analytic argument; here $H_j$ is the $j$th harmonic number. Since $H_{k-2}\le1+\ln k$, this lowers the leading coefficient of $\ln k$ from $5$ to $2.108$. We also determine $R_2=5$ and $R_3=185/32$ exactly; both are suprema that are approached but not attained. On the lower side we prove $R_k\ge2(1+H_{k-1})$ for every $k$, and give a second family of lower bounds that is larger than this for $2\le k\le18$. Previous analyses bound the expected cost of a cluster just after it is first sampled by a constant times its optimal cost, and then charge that cost for the rest of the run. We instead track how a cluster's cost evolves after its first hit: a cluster left with a large residual cost is likely to be sampled again before it accumulates much of the logarithmic charge, and each repeat repairs it.

Authors: Aravind Reddy

Let $R_k$ be the worst-case ratio between the expected cost of the $k$ seeds chosen by $k$-means++ and the optimal $k$-means cost. In their seminal paper introducing $k$-means++, Arthur and Vassilvitskii (SODA 2007) showed that $2\ln k\le R_k\le8(\ln k+2)$, and Makarychev, Reddy and Shan (NeurIPS 2020) improved the upper bound to $5(\ln k+2)$. We prove $R_k\le\frac{1163}{233}+2.108\,H_{k-2}$ for every $k\ge3$ by a computer-assisted argument, and $R_k\le5+2.3139\,H_{k-2}$ for every $k\ge2$ by a short analytic argument; here $H_j$ is the $j$th harmonic number. Since $H_{k-2}\le1+\ln k$, this lowers the leading coefficient of $\ln k$ from $5$ to $2.108$. We also determine $R_2=5$ and $R_3=185/32$ exactly; both are suprema that are approached but not attained. On the lower side we prove $R_k\ge2(1+H_{k-1})$ for every $k$, and give a second family of lower bounds that is larger than this for $2\le k\le18$. Previous analyses bound the expected cost of a cluster just after it is first sampled by a constant times its optimal cost, and then charge that cost for the rest of the run. We instead track how a cluster's cost evolves after its first hit: a cluster left with a large residual cost is likely to be sampled again before it accumulates much of the logarithmic charge, and each repeat repairs it.

Wednesday, November 11 2026, 01:00

Dana’s talk on OpenAI’s proof of the Unique Games Conjecture, now on YouTube!

from Scott Aaronson

Enjoy all three hours of this! Dana covers what the Unique Games problem is and why people cared about it, the structure of a proof of its NP-hardness, exactly where the previous attempts to complete the proof (based on the Long Code and the Hadamard Code) got stuck, the new tree-based code that the AI […]

Enjoy all three hours of this! Dana covers what the Unique Games problem is and why people cared about it, the structure of a proof of its NP-hardness, exactly where the previous attempts to complete the proof (based on the Long Code and the Hadamard Code) got stuck, the new tree-based code that the AI discovered, and how the new code gets around the Long Code and Hadamard Code’s limitations. She doesn’t get to the soundness analysis.

Feel free to ask questions in the comments. I might write some notes here later with my own doofus understanding of Dana’s talk. Experts are of course hard at work on more formal writeups of the proof (which will hopefully be more comprehensible than the AI’s writeup).

By Scott

Tuesday, November 10 2026, 20:27

TCS+ talk: Wednesday, November 18 — Sahil Singla, Georgia Tech

from TCS+ Seminar Series

The next TCS+ talk will take place next Wednesday, November 18th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Sahil Singla from Georgia Tech will discuss recent developments on “The Matroid Secretary Problem” (abstract below). You can reserve a spot as an individual or a group to join […]

The next TCS+ talk will take place next Wednesday, November 18th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Sahil Singla from Georgia Tech will discuss recent developments on “The Matroid Secretary Problem” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Registration is not required to attend the interactive talk, and the link will be posted on the website the day prior to the talk; however, by registering in the form, you will receive a reminder, along with the link. (The recorded talk will also be posted on our website afterwards) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Abstract: In the classic secretary problem, n unknown values are revealed in a uniformly random order, and the goal is to select the maximum. We may select at most one value, and each decision must be immediate and irrevocable. A simple threshold rule selects the maximum with probability at least 1/e. What if we want to select multiple values to maximize their sum, subject to a feasibility constraint? Matroids capture many natural constraints, such as selecting at most k elements or selecting a set of edges with no cycle. In 2007, Babaioff, Immorlica, and Kleinberg conjectured that for every matroid constraint, an online algorithm can obtain a constant fraction of the hindsight optimum value in expectation. In this talk, I will discuss progress on this problem over the past two decades, culminating in the recent resolution of the conjecture in its strong form: achieving a 1/e fraction of the hindsight optimum in expectation, which is best possible even for selecting a single value.

Based on joint work with Hamed Abdi, Kiarash Banihashem, MohammadTaghi Hajiaghayi, Zhiyi Huang, and Danny Mittal, available at arXiv:2611.08869.

By plustcs

Tuesday, November 10 2026, 19:33

nothing left to prove song, and a ripoff of my heroic age post?

from Emanuele Viola

I was pointed to this amusing, but nowadays completely banal, song, and my trained eye spotted a frame that seems to be taken from my previous blog post the heroic age of mathematical exploration: circa 3000 BCE — 2026 CE:

By Manu

Tuesday, November 10 2026, 18:45

IHP school on AI for mathematicians and theoretical physicists

from CS Theory Events

February 22-26, 2027 Institut Henri Poincare, Paris indico.math.cnrs.fr/event/17079/ Registration deadline: November 30, 2026 Recent breakthroughs have shown that AI can now prove theorems. At the same time, many of the deepest open questions in AI are, at heart, questions of mathematics and physics: high-dimensional geometry, non-convex optimisation, the statistical mechanics of learning. AI is becoming … Continue reading IHP school on AI for mathematicians and theoretical physicists

By shacharlovett

February 22-26, 2027 Institut Henri Poincare, Paris https://indico.math.cnrs.fr/event/17079/ Registration deadline: November 30, 2026 Recent breakthroughs have shown that AI can now prove theorems. At the same time, many of the deepest open questions in AI are, at heart, questions of mathematics and physics: high-dimensional geometry, non-convex optimisation, the statistical mechanics of learning. AI is becoming … Continue reading IHP school on AI for mathematicians and theoretical physicists

By shacharlovett

Tuesday, November 10 2026, 17:29

No One Buys Corn in the Corn Market

from Ben Recht

Prediction markets are for gambling. But what part of finance isn’t gambling?

Hi there, argmin readers. Today’s post is a live blog of Class 19 of my graduate seminar “Forecasting: A Critical Retrospective.” In this class, we try to make sense of prediction markets. Good luck to us! The syllabus and list of past posts are here.

On May 16, 2008, in the Policy Forum of Science magazine, two dozen notable scholars called 1 on the Commodity Futures Trading Commission (CFTC) to create a safe harbor for prediction markets. In their view, “vibrant, liquid prediction markets” would improve decision-making by “helping businesses make better investment decisions,” “helping governments make better fiscal and monetary policy decisions,” and helping “private firms and public institutions in managing economic risks.” The predictive magic came from the unreasonable efficiency of markets:

“Because information is often widely dispersed among economic actors, it is highly desirable to find a mechanism to collect and aggregate that information. Free markets usually manage this process well because almost anyone can participate, and the potential for profit (and loss) creates strong incentives to search for better information.”

Just two months earlier, on March 17, 2008, the Federal Reserve had convened an emergency meeting, its first in thirty years, to guarantee the failing loans of Bear Stearns and prevent it from collapsing into bankruptcy. The world would quickly plummet into the worst economic recession since the Great Depression. It was the efficiency of markets in action.

Fast forward to 2026, and prediction markets have parasitically clawed their way into ubiquity. Trading volume on sketchy legalized gambling sites Polymarket and Kalshi grew fivefold this year. These companies are also aggressively lobbying the CFTC, but without the technocratic idealism of the 2008 policy wonks. Kalshi and Polymarket are gambling websites, and everyone knows it. But they are using old academic arguments to skirt state gambling laws. The CFTC regulates futures markets. The states regulate gambling. If the CFTC says prediction markets are futures markets, then they are protected from the authority of the states. The Trump family is heavily invested in crypto and prediction markets and is pushing hard to bend the CFTC to their will to enshrine the legal status of the companies.2

The CFTC and its regulatory predecessors have always been tasked with carving out “safe harbors” for futures markets because you have to tie yourself into neoliberal knots to explain how derivatives markets are not gambling. Like most financial instruments, futures markets rest on a simple idea that feels reasonable, but it spirals out of control once you let the invisible hand of the market get to work.

A company that relies on corn production might want to set a price for corn in advance so they can manage risks from volatile harvests. They make a contract with some farm producers before the season, agreeing upon how much they’ll pay per bushel at the end of the season. If there’s a rough harvest, the market price will be considerably higher than the contract price, and the buyer gets their corn at a discount. On the other hand, if the corn harvest is plentiful, the contract price will be higher than the market rate. However, this isn’t the end of the world, because they can buy corn at a cheaper rate from another farmer as supply is abundant. Properly balancing corn on futures contracts and corn at actual prices leads to an expected cost the buyer feels comfortable with. The futures contract serves as a risk management strategy, allowing producers and buyers to hedge against an uncertain market.

Now, a market fundamentalist would argue that for everyone to get the best prices for their businesses, a bigger, more liquid market should be established to determine the right price. People should be able to sell their contracts to each other to dial in the right price as new information emerges. But this ideal world of making all sorts of contracts with different producers and buyers is physically messy. If you wanted to execute all of these deals, you’d have to ship the corn all over the place at harvest time, creating a logistical nightmare. So instead, people created a market for these futures, where you’d sell your futures contract before it ever gets executed. No corn moves anywhere, but money still changes hands. The risk management stuff still works out for all involved, but no one is buying any actual corn in the corn market.

And shoot, if you’re not buying actual corn, why not buy virtual everything else too in these markets? You’re buying a contract on the price of corn, not the corn. You could imagine buying a contract on the price of a financial instrument, like a stock price or an interest rate. This would help you manage risk in an investment portfolio. After its establishment in the 1970s, the CFTC began liberally including all sorts of futures in its “no, this is actually not gambling” bucket. Sure, you’re not buying anything in these markets except contracts on future prices. But it was of economic value to allow such speculation. The CFTC still discouraged derivative contracts on “events” that didn’t have a clear economic purpose.

Meanwhile, egghead economists argued the CFTC should relax. Any event can be securitized, and this securitization is good for everyone. Ken Arrow and Gerard Debreu won the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for this sort of thinking. Inspired, economists in Iowa started running fake markets to predict the outcome of the Iowa caucuses. And people like Phil Tetlock started to run competitions with IARPA to improve forecasting geopolitical turmoil. But prediction markets add another layer of derivatives. Betting on events lets decision makers discover probabilities they can plug into expected-value calculations, so they know how to price commodity futures contracts. I guess?

I am personally fascinated by this pattern of derivatization in finance. Derivatives feel sensible for risk management, but require creating a bizarre abstraction of the object you’d like to hedge against. These derivatives aren’t necessarily gambling, but derivatives on derivatives are always gambling. There’s no way to avoid the cascade into speculation once you financialize reality.

Prediction markets are a hilarious caricature of neoliberal economics. In prediction markets, everything is gambling, and everyone knows it. No matter what the rationalist disciples of Tetlock tell you, there’s no sensible defense of the prediction markets. They don’t produce probabilities in any reasonable sense, in theory or in practice. They don’t inform decision-making. They are not “efficient,” as no markets are efficient. The prediction market companies exist entirely for gambling. And gamblers love them.

If you want to defend them, you need to embrace a certain style of hyper-rational thinking that we’ve mistakenly normalized. We are told that the predictions are good because markets are the ultimate information aggregators, and the information is good because people in markets have skin in the game. But who takes advice from degenerate gamblers?

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1

I mean, look at this author list. The argmin nightmare blunt rotation. Kenneth Arrow, Robert Forsythe, Michael Gorham, Robert Hahn, Robin Hanson, John Ledyard, Saul Levmore, Robert Litan, Paul Milgrom, Forrest D. Nelson, George R. Neumann, Marco Ottaviani, Thomas C. Schelling, Robert J. Shiller, Vernon L. Smith, Erik Snowberg, Cass Sunstein, Paul Tetlock, Philip Tetlock, Hal R. Varian, Justin Wolfers, Eric Zitzewitz.

2

For more on these developments, read John Cassidy’s reporting “This Is What Trumpian Self-Dealing Looks Like” in The New Yorker.

By Ben Recht

Tuesday, November 10 2026, 15:20

TR26-316 | Hard Instances for Strong Semantic Proof Systems: CNFs for Depth-Four PC and Deep Instances for AC0[2]-Frege | Yaroslav Alekseev, Svyatoslav Gryaznov, Dmitry Itsykson, Toniann Pitassi, Robert Robere, Iddo Tzameret

from ECCC Papers

We establish a collection of new size lower bounds against fairly strong algebraic-Boolean propositional proof systems, from depth-3 semantic polynomial calculus for CNFs, up to semantic $AC^0[2]$-Frege in the regime of deep formulas, as follows: (1) $\Sigma\Pi\Sigma$-PC: Exponential lower bound against semantic $\Sigma\Pi\Sigma$-PC over $F_2$, namely, polynomial calculus operating with depth-3 circuits with sum top gate, for the bit pigeonhole principle. We further obtain $2^{\Omega(n)}$ lower bounds for random 3-CNFs on $n$ variables and for the bit and functional pigeonhole principles with $n+1$ pigeons and $n$ holes. (2) $\Sigma\Pi\Sigma\Pi$-PC: Superpolynomial lower bounds for semantic $\Sigma\Pi\Sigma\Pi$-PC over $F_2$, with unrestricted bottom fan-in. The hard instances include the weak unary pigeonhole principle and lifting of random CNFs of logarithmic width. For the weak pigeonhole principle with $2n$ pigeons and $n$ holes, the lower bound is $\exp(\Omega((\log n)^2/\log\log n))$. This further settles the open problem of establishing superpolynomial lower bounds against $\mathrm{Res}(\log)$ for this principle. When the $\Sigma\Pi\Sigma\Pi$-PC bottom fan-in is logarithmic, we obtain exponential lower bounds for random CNFs of logarithmic width. (3) Depth-$d$ $AC^0[2]$-Frege: For every fixed $d\ge 2$, we construct unsatisfiable families of formulas of depth $d$ that require superpolynomial-size semantic depth-$d$ $AC^0[2]$-Frege refutations, but have polynomial-size semantic depth-$(d+1)$ $AC^0[2]$-Frege refutations. This provides the first unconditional hard instances against arbitrary depth $AC^0[2]$-Frege. The arguments build and expand on the recent AI-generated proof of an exponential lower bound against resolution over parities by Braun [16]. We distil Braun's implicit annihilating polynomial argument into an abstract framework based on standard pseudo-reduction operators, and augment it with new techniques. We show how rank lower bounds for these operators imply proof-size lower bounds. The polynomial approximation method of Razborov and Smolensky [60, 64] extends the framework to bounded bottom degree. Lifting and random restrictions remove the restriction on bottom fan-in. For the $AC^0[2]$-Frege result, we combine polynomial approximation with the common annihilating polynomial and a lifting construction based on constant-depth formulas for the coin problem.

Tuesday, November 10 2026, 04:11

A short proof of the Aaronson-Ambainis Conjecture via the completion framework

Authors: Thomas Huffstutler, David Miloschewsky, Supartha Podder

We give a short proof of the Aaronson-Ambainis conjecture. If a partial Boolean function $f$ is defined on a $δ$-fraction of all $n$-bit strings and have quantum query complexity $Q(f)$, then, for every $0<ε<1$, a deterministic algorithm computes $f$ correctly on at least a $1-ε$ fraction of its domain using $O( Q(f)^6 / (εδ)^2)$ queries. Our proof uses the completion framework and establishes a more general bound in terms of the classical adversary measure. The only result we use from Turturean's recent proof is a special case of his certificate inequality, for which we provide a self-contained proof. We also derive the stronger variant of the Aaronson-Ambainis Conjecture, the bounded-polynomial influence conjecture, from our main result in the appendix.

Authors: Thomas Huffstutler, David Miloschewsky, Supartha Podder

We give a short proof of the Aaronson-Ambainis conjecture. If a partial Boolean function $f$ is defined on a $δ$-fraction of all $n$-bit strings and have quantum query complexity $Q(f)$, then, for every $0<ε<1$, a deterministic algorithm computes $f$ correctly on at least a $1-ε$ fraction of its domain using $O( Q(f)^6 / (εδ)^2)$ queries. Our proof uses the completion framework and establishes a more general bound in terms of the classical adversary measure. The only result we use from Turturean's recent proof is a special case of his certificate inequality, for which we provide a self-contained proof. We also derive the stronger variant of the Aaronson-Ambainis Conjecture, the bounded-polynomial influence conjecture, from our main result in the appendix.

Tuesday, November 10 2026, 01:00

Succinct Infinite-Horizon Lyapunov and LQR Value Problems Are BQP-Complete

Authors: Mohamed Serry, Jun Liu

Fundamental tasks in linear systems analysis and control require evaluating infinite-horizon output energies for autonomous systems and optimal value functions for controlled systems. These quantities are characterized by Lyapunov and algebraic Riccati equations, respectively. In this work, we study promise problems for their evaluation when the system matrices are sparse and represented succinctly by Boolean circuits specifying their nonzero entries, with input length polynomial in the logarithm of the system dimension. For dissipative autonomous systems with coordinate-vector initial states, we define the succinct Lyapunov value problem (SLV). Its extension to linear-quadratic optimal control is the succinct LQR value problem (SLQRV). We prove that SLV is $\BQP$-hard: every problem in bounded-error quantum polynomial time ($\BQP$) reduces to it in polynomial time. We then establish that the more general SLQRV problem admits a quantum algorithm running in time polynomial in the succinct input length. Since SLV is a special case of SLQRV, both problems are $\BQP$-complete. Consequently, unless bounded-error probabilistic polynomial time coincides with $\BQP$, no classical randomized algorithm can solve either problem on all promised instances in time polynomial in the succinct input length.

Authors: Mohamed Serry, Jun Liu

Fundamental tasks in linear systems analysis and control require evaluating infinite-horizon output energies for autonomous systems and optimal value functions for controlled systems. These quantities are characterized by Lyapunov and algebraic Riccati equations, respectively. In this work, we study promise problems for their evaluation when the system matrices are sparse and represented succinctly by Boolean circuits specifying their nonzero entries, with input length polynomial in the logarithm of the system dimension. For dissipative autonomous systems with coordinate-vector initial states, we define the succinct Lyapunov value problem (SLV). Its extension to linear-quadratic optimal control is the succinct LQR value problem (SLQRV). We prove that SLV is $\BQP$-hard: every problem in bounded-error quantum polynomial time ($\BQP$) reduces to it in polynomial time. We then establish that the more general SLQRV problem admits a quantum algorithm running in time polynomial in the succinct input length. Since SLV is a special case of SLQRV, both problems are $\BQP$-complete. Consequently, unless bounded-error probabilistic polynomial time coincides with $\BQP$, no classical randomized algorithm can solve either problem on all promised instances in time polynomial in the succinct input length.

Tuesday, November 10 2026, 01:00

$\mathsf{EQP}\subseteq\mathsf{IP}$ Does Not Relativize

Authors: Sinan Oral, Supartha Podder

We construct a recursive oracle $O$ and a unary language $L\in\mathsf{EQP}^O\setminus(\mathsf{IP}^O\cup\mathsf{PH}^O)$, strengthening the $\mathsf{BQP}$ versus $\mathsf{IP}$ oracle separation of Bouland et al. to exact quantum computation. The separation is based on a variant of the Forrelation problem in which the normalized correlation of one Boolean function with the Fourier transform of another is promised to be exactly $+1/2$ or $-1/2$. The main technical contribution is a repair method that constructs distributions supported on these exact levels while preserving the indistinguishability needed for the $\mathsf{IP}$ and $\mathsf{PH}$ lower bounds. A uniform polynomial-time quantum algorithm distinguishes the two cases with certainty using only three oracle queries.

Authors: Sinan Oral, Supartha Podder

We construct a recursive oracle $O$ and a unary language $L\in\mathsf{EQP}^O\setminus(\mathsf{IP}^O\cup\mathsf{PH}^O)$, strengthening the $\mathsf{BQP}$ versus $\mathsf{IP}$ oracle separation of Bouland et al. to exact quantum computation. The separation is based on a variant of the Forrelation problem in which the normalized correlation of one Boolean function with the Fourier transform of another is promised to be exactly $+1/2$ or $-1/2$. The main technical contribution is a repair method that constructs distributions supported on these exact levels while preserving the indistinguishability needed for the $\mathsf{IP}$ and $\mathsf{PH}$ lower bounds. A uniform polynomial-time quantum algorithm distinguishes the two cases with certainty using only three oracle queries.

Tuesday, November 10 2026, 01:00

Type-logical grammars and formal languages

Authors: Richard Moot

We show that type-logical grammars with the standard rules for product, associativity, extraction and infixation have an NP-complete fixed recognition problem, adapting a construction for first-order linear logic from Pshenitsyn (2025), who already showed that grammars in first-order linear logic are closed under intersection and therefore not semilinear and not mildly context-sensitive. We strengthen these results and shown that grammars in first-order linear logic can simulate the quasi-linear grammars of Book \& Greibach (1970), the grammars recognised in non-deterministic linear time (NSPACE(n) or. NLIN). I argue that NLIN is an interesting class of formal languages as well, and a useful upper bound on the time complexity of formalisms used for natural language, just like LOGCFL is a useful upper bound on the space complexity. Taken together this means that first-order linear logic can be seen as the logic of non-deterministic linear-time string languages, graph languages and automata.

Authors: Richard Moot

We show that type-logical grammars with the standard rules for product, associativity, extraction and infixation have an NP-complete fixed recognition problem, adapting a construction for first-order linear logic from Pshenitsyn (2025), who already showed that grammars in first-order linear logic are closed under intersection and therefore not semilinear and not mildly context-sensitive. We strengthen these results and shown that grammars in first-order linear logic can simulate the quasi-linear grammars of Book \& Greibach (1970), the grammars recognised in non-deterministic linear time (NSPACE(n) or. NLIN). I argue that NLIN is an interesting class of formal languages as well, and a useful upper bound on the time complexity of formalisms used for natural language, just like LOGCFL is a useful upper bound on the space complexity. Taken together this means that first-order linear logic can be seen as the logic of non-deterministic linear-time string languages, graph languages and automata.

Tuesday, November 10 2026, 01:00

Polynomial Sample Complexity of Noninteractive Simulation over Finite Alphabets

Authors: Haoran Wang

We study noninteractive simulation from a finite correlated source. Alice and Bob observe independent copies of the source and generate outputs without communication. We give a quantitative construction for arbitrary finite output alphabets: for every fixed source, a joint law that is simulable with arbitrarily many samples can be approximated using a number of samples polynomial jointly in the output alphabet sizes and the inverse accuracy. We first study binary outputs. A one-sided Wright--Fisher comparison gives a source-dependent explicit exponent and a finite-head Gaussian-threshold representation. We then handle arbitrary finite output alphabets through averaged influence regularity, simplex-preserving Gaussian moment lifting, and a common Gaussian projection for all correlated-prefix states. The sample exponent is independent of the output alphabet sizes. For a concrete biased binary source we certify an explicit exponent below $19$.

Authors: Haoran Wang

We study noninteractive simulation from a finite correlated source. Alice and Bob observe independent copies of the source and generate outputs without communication. We give a quantitative construction for arbitrary finite output alphabets: for every fixed source, a joint law that is simulable with arbitrarily many samples can be approximated using a number of samples polynomial jointly in the output alphabet sizes and the inverse accuracy. We first study binary outputs. A one-sided Wright--Fisher comparison gives a source-dependent explicit exponent and a finite-head Gaussian-threshold representation. We then handle arbitrary finite output alphabets through averaged influence regularity, simplex-preserving Gaussian moment lifting, and a common Gaussian projection for all correlated-prefix states. The sample exponent is independent of the output alphabet sizes. For a concrete biased binary source we certify an explicit exponent below $19$.

Tuesday, November 10 2026, 01:00

On Non-Deterministic Representation of Boolean Functions

Authors: Sudharshan Rajagopalan, Nitin Saurabh

We study non-deterministic representation of Boolean functions by polynomials, introduced by (de Wolf, 2000). A polynomial $p(x)$ is said to be a non-deterministic polynomial for a Boolean function $f$ if for all $a\in\{0,1\}^n$, $p(a)\neq 0$ iff $f(a)=1$. The non-deterministic degree of $f$, denoted $\mathsf{ndeg}(f)$, is the minimal degree of a non-deterministic polynomial for $f$. In this work, we show the following main results: - Almost all Boolean functions on $n$ variables have non-deterministic degree at most $n/2 + O(\sqrt{n\log n})$. This confirms a conjecture of (Iyer et al., 2023). Together with the lower bound of (Iyer et al., 2023) it establishes a tight bound on the rational degree of a random Boolean function up to lower order terms. - We disprove a conjecture of (Kothari et al., 2026) about effective hypercube nullstellensatz for more than two polynomials. On the positive side, we also identify a constraint under which an effective hypercube nullstellensatz holds for many polynomials.

Authors: Sudharshan Rajagopalan, Nitin Saurabh

We study non-deterministic representation of Boolean functions by polynomials, introduced by (de Wolf, 2000). A polynomial $p(x)$ is said to be a non-deterministic polynomial for a Boolean function $f$ if for all $a\in\{0,1\}^n$, $p(a)\neq 0$ iff $f(a)=1$. The non-deterministic degree of $f$, denoted $\mathsf{ndeg}(f)$, is the minimal degree of a non-deterministic polynomial for $f$. In this work, we show the following main results: - Almost all Boolean functions on $n$ variables have non-deterministic degree at most $n/2 + O(\sqrt{n\log n})$. This confirms a conjecture of (Iyer et al., 2023). Together with the lower bound of (Iyer et al., 2023) it establishes a tight bound on the rational degree of a random Boolean function up to lower order terms. - We disprove a conjecture of (Kothari et al., 2026) about effective hypercube nullstellensatz for more than two polynomials. On the positive side, we also identify a constraint under which an effective hypercube nullstellensatz holds for many polynomials.

Tuesday, November 10 2026, 01:00

Simulating Nondeterministic Turing Machines in Subexponential Deterministic Time

Authors: Carmen Strassle, Bernardo Subercaseaux

We show that every multitape nondeterministic Turing machine can be simulated deterministically in subexponential time. Specifically, we show that the recent algorithms for $3$-$\mathsf{SAT}$ presented by OpenAI and Anthropic imply $\mathsf{NTIME}[t]\subseteq\mathsf{DTIME}[2^{O(t \log \log t/\sqrt{\log t})}]$, and that a stronger consequence of their proof yields \[\mathsf{NTIME}[t] \subseteq\mathsf{DTIME}[2^{O(t /\sqrt{\log t})}]. \] More generally, we show that any subexponential time algorithm for solving circuit-CSPs transfers to a deterministic simulation of $\mathsf{NTIME}$ with the same savings. Our results follow from new generalizations of Cook--Levin, which may be of independent interest.

Authors: Carmen Strassle, Bernardo Subercaseaux

We show that every multitape nondeterministic Turing machine can be simulated deterministically in subexponential time. Specifically, we show that the recent algorithms for $3$-$\mathsf{SAT}$ presented by OpenAI and Anthropic imply $\mathsf{NTIME}[t]\subseteq\mathsf{DTIME}[2^{O(t \log \log t/\sqrt{\log t})}]$, and that a stronger consequence of their proof yields \[\mathsf{NTIME}[t] \subseteq\mathsf{DTIME}[2^{O(t /\sqrt{\log t})}]. \] More generally, we show that any subexponential time algorithm for solving circuit-CSPs transfers to a deterministic simulation of $\mathsf{NTIME}$ with the same savings. Our results follow from new generalizations of Cook--Levin, which may be of independent interest.

Tuesday, November 10 2026, 01:00

Proof of the Aaronson-Ambainis Conjecture: An Exposition

Authors: Guy Blanc, Jordan Docter, Carmen Strassle, Li-Yang Tan

We give an exposition of a recent proof of the Aaronson-Ambainis conjecture, which states that every degree-$d$ polynomial $p : \{0,1\}^n \to [0,1]$ has a variable of influence at least $\mathrm{poly}(\mathrm{Var}(p)/d)$. This implies that quantum query algorithms for decision problems can be efficiently simulated classically on most inputs. It also yields a near-equivalence between the status of the $\mathsf{BPP}$ vs. $\mathsf{BQP}$ problem in the random oracle and unrelativized worlds. We organize the proof around four main steps and discuss how each connects to the literature on the analysis of boolean functions and probabilistic combinatorics. To bring out the key ideas, we do not optimize parameters, and our presentation yields weaker exponents than the original. $\textbf{Authors' note.}$ The proof was discovered by David Turturean using a research system developed at Physical Superintelligence (PSI), in combination with a non-public Anthropic model. The proof was then shared with the authors, who worked to understand it and prepare this exposition. Anthropic's public models were used in this process. The authors did not receive financial compensation from Anthropic or PSI. This article is not intended for publication at a conference or in a journal.

Authors: Guy Blanc, Jordan Docter, Carmen Strassle, Li-Yang Tan

We give an exposition of a recent proof of the Aaronson-Ambainis conjecture, which states that every degree-$d$ polynomial $p : \{0,1\}^n \to [0,1]$ has a variable of influence at least $\mathrm{poly}(\mathrm{Var}(p)/d)$. This implies that quantum query algorithms for decision problems can be efficiently simulated classically on most inputs. It also yields a near-equivalence between the status of the $\mathsf{BPP}$ vs. $\mathsf{BQP}$ problem in the random oracle and unrelativized worlds. We organize the proof around four main steps and discuss how each connects to the literature on the analysis of boolean functions and probabilistic combinatorics. To bring out the key ideas, we do not optimize parameters, and our presentation yields weaker exponents than the original. $\textbf{Authors' note.}$ The proof was discovered by David Turturean using a research system developed at Physical Superintelligence (PSI), in combination with a non-public Anthropic model. The proof was then shared with the authors, who worked to understand it and prepare this exposition. Anthropic's public models were used in this process. The authors did not receive financial compensation from Anthropic or PSI. This article is not intended for publication at a conference or in a journal.

Tuesday, November 10 2026, 01:00

Closing the Complexity Gap for Wheeler Graph Recognition on $d$-NFAs

Authors: Kengo Hashimoto

Wheeler graph recognition asks whether a directed graph whose arcs are labeled by symbols from a totally ordered alphabet admits a Wheeler order, a total order on its vertices compatible with the labels of arcs and the order of their tails. A $d$-NFA is a nondeterministic finite automaton in which at most $d$ transitions with any given label leave each state. For Wheeler graph recognition on $d$-NFAs, Alanko et al. gave a polynomial-time algorithm for $d \leq 2$, whereas Gibney and Thankachan proved NP-completeness for every fixed $d \geq 5$. The computational complexity for $d = 3$ and $d = 4$ remained open. We resolve both open cases by extending the reduction of Gibney and Thankachan with a new encoding of clauses of Not-All-Equal 3-SAT. Specifically, we prove that recognition is NP-complete even for directed acyclic graphs over an ordered binary alphabet with a unique source reaching every vertex and total out-degree at most three at every vertex. Consequently, Wheeler graph recognition on $d$-NFAs is NP-complete for every fixed $d \geq 3$.

Authors: Kengo Hashimoto

Wheeler graph recognition asks whether a directed graph whose arcs are labeled by symbols from a totally ordered alphabet admits a Wheeler order, a total order on its vertices compatible with the labels of arcs and the order of their tails. A $d$-NFA is a nondeterministic finite automaton in which at most $d$ transitions with any given label leave each state. For Wheeler graph recognition on $d$-NFAs, Alanko et al. gave a polynomial-time algorithm for $d \leq 2$, whereas Gibney and Thankachan proved NP-completeness for every fixed $d \geq 5$. The computational complexity for $d = 3$ and $d = 4$ remained open. We resolve both open cases by extending the reduction of Gibney and Thankachan with a new encoding of clauses of Not-All-Equal 3-SAT. Specifically, we prove that recognition is NP-complete even for directed acyclic graphs over an ordered binary alphabet with a unique source reaching every vertex and total out-degree at most three at every vertex. Consequently, Wheeler graph recognition on $d$-NFAs is NP-complete for every fixed $d \geq 3$.

Tuesday, November 10 2026, 01:00

A note on exact external information complexity

Authors: Henning Wunderlich

We study the prior-free zero-error private-coin external information complexity $B(f)$ of finite two-party functions, with correctness on every input. An elementary rectangle/entropy argument gives $B(\mathsf{AND})=\log_2 3$. A complete classification of Boolean $3\times3$ matrices leaves, up to input permutations and output complementation, only the two-overlapping-rectangles function $F$ without an exact value. An entropy potential and a finite rational protocol give $2.1212661667060486842\ldots\le B(F)<2.130998318$. We also determine its deterministic external information and zero-error communication costs. A separate product-prior certificate and internal-information additivity yield $R_0^{\mathrm{pub}}(F^{\times t})\ge2.021991187178562\ldots\,t$, strictly exceeding the fractional-cover rate $2t$.

Authors: Henning Wunderlich

We study the prior-free zero-error private-coin external information complexity $B(f)$ of finite two-party functions, with correctness on every input. An elementary rectangle/entropy argument gives $B(\mathsf{AND})=\log_2 3$. A complete classification of Boolean $3\times3$ matrices leaves, up to input permutations and output complementation, only the two-overlapping-rectangles function $F$ without an exact value. An entropy potential and a finite rational protocol give $2.1212661667060486842\ldots\le B(F)<2.130998318$. We also determine its deterministic external information and zero-error communication costs. A separate product-prior certificate and internal-information additivity yield $R_0^{\mathrm{pub}}(F^{\times t})\ge2.021991187178562\ldots\,t$, strictly exceeding the fractional-cover rate $2t$.

Tuesday, November 10 2026, 01:00

On the Complexity of Extending Prescribed Linear Forests to Hamiltonian Cycles in Hypercubes

Authors: Huazhong Lü, Tingzeng Wu

Let $Q_n$ be the $n$-dimensional hypercube, and let $P\subseteq E(Q_n)$ be a set of prescribed edges. It is known that, for $n\ge 2$, every such set $P$ with $|P|\leq 2n-3$ whose edges form a linear forest is contained in a Hamiltonian cycle of $Q_n$. This bound is sharp for general linear forests. The computational complexity of deciding whether a prescribed edge set of a hypercube can be extended to a Hamiltonian cycle was posed as an open question by Dvořák [SIAM J. Discrete Math. 19 (2005) 135--144]. We prove that the problem is NP-complete even when the prescribed edges form a spanning linear forest with no isolated vertices. We give a polynomial-time reduction from a restricted 3-SAT problem, using local spanning path-forest gadgets for variables and clauses that are uniformly embedded in the hypercube. The same reduction therefore establishes NP-hardness of two versions of Hamiltonian-cycle extension problems for general prescribed edge sets.

Authors: Huazhong Lü, Tingzeng Wu

Let $Q_n$ be the $n$-dimensional hypercube, and let $P\subseteq E(Q_n)$ be a set of prescribed edges. It is known that, for $n\ge 2$, every such set $P$ with $|P|\leq 2n-3$ whose edges form a linear forest is contained in a Hamiltonian cycle of $Q_n$. This bound is sharp for general linear forests. The computational complexity of deciding whether a prescribed edge set of a hypercube can be extended to a Hamiltonian cycle was posed as an open question by Dvořák [SIAM J. Discrete Math. 19 (2005) 135--144]. We prove that the problem is NP-complete even when the prescribed edges form a spanning linear forest with no isolated vertices. We give a polynomial-time reduction from a restricted 3-SAT problem, using local spanning path-forest gadgets for variables and clauses that are uniformly embedded in the hypercube. The same reduction therefore establishes NP-hardness of two versions of Hamiltonian-cycle extension problems for general prescribed edge sets.

Tuesday, November 10 2026, 01:00

Linear Objective Problems: A Complexity Framework for Bilevel Optimization

Authors: Christoph Grüne, James B. Orlin, Lasse Wulf

Many bilevel and robust optimization problems are complete for the second level of the polynomial hierarchy, but such results are usually proven one problem at a time with problem-specific reductions. A recurring obstacle is that the lower level must select an optimal solution rather than merely a feasible one, whereas standard reductions preserve only whether a solution exists, not which solutions are optimal. We introduce linear objective problems (LOP), a framework for optimization problems whose solutions are explicit subsets of a ground set and whose objective is linear in these elements. We define tight solution-embedding reductions, which map the optimal solutions of one problem element by element onto the optimal solutions of another, without having to recognize optimal solutions, which is coNP-hard in general. This gives a notion of LOP-completeness, with Satisfiability as the canonical complete problem, even with unit weights. We show that many classical problems are LOP-complete, including Vertex Cover, Dominating Set, TSP, Knapsack, and Steiner Tree, and we introduce a duality that transfers LOP-completeness between minimization and maximization problems. As our main application, we prove a meta-theorem for min-max regret optimization under interval uncertainty: for every LOP-complete problem, the min-max regret version is $Σ_2^p$-complete. Instead of a separate proof for each nominal problem, it suffices to show that the nominal problem is LOP-complete. This recovers known results for the interval min-max regret versions of Knapsack and Maximum Benefit Set Cover and yields completeness for over 50 further problems.

Authors: Christoph Grüne, James B. Orlin, Lasse Wulf

Many bilevel and robust optimization problems are complete for the second level of the polynomial hierarchy, but such results are usually proven one problem at a time with problem-specific reductions. A recurring obstacle is that the lower level must select an optimal solution rather than merely a feasible one, whereas standard reductions preserve only whether a solution exists, not which solutions are optimal. We introduce linear objective problems (LOP), a framework for optimization problems whose solutions are explicit subsets of a ground set and whose objective is linear in these elements. We define tight solution-embedding reductions, which map the optimal solutions of one problem element by element onto the optimal solutions of another, without having to recognize optimal solutions, which is coNP-hard in general. This gives a notion of LOP-completeness, with Satisfiability as the canonical complete problem, even with unit weights. We show that many classical problems are LOP-complete, including Vertex Cover, Dominating Set, TSP, Knapsack, and Steiner Tree, and we introduce a duality that transfers LOP-completeness between minimization and maximization problems. As our main application, we prove a meta-theorem for min-max regret optimization under interval uncertainty: for every LOP-complete problem, the min-max regret version is $Σ_2^p$-complete. Instead of a separate proof for each nominal problem, it suffices to show that the nominal problem is LOP-complete. This recovers known results for the interval min-max regret versions of Knapsack and Maximum Benefit Set Cover and yields completeness for over 50 further problems.

Tuesday, November 10 2026, 01:00

Do Feature Interactions Break Explainability? Provable Explanations for Any-Order Neural Additive Models

Authors: Idan Refaeli, Yizhak Yisrael Elboher, Guy Katz, Shahaf Bassan

Post-hoc explanation methods for neural networks are often heuristic and lack provable guarantees. A common alternative to such methods that was studied in recent years is to compute a cardinally-minimal subset of input features that is provably sufficient to determine the model's prediction. For general neural networks, however, finding such explanations is computationally intractable. In contrast, recent work showed that this problem becomes more tractable for a particular family of neural networks, namely neural additive models (NAMs), though standard NAMs are highly restrictive because they exclude feature interactions altogether. In this work, we study the computation of provably minimal and sufficient explanations for neural networks with restricted feature interactions, bridging the gap between fully general neural networks and purely additive models. We first prove that even with constant-order interactions, the problem is $Σ_2^P$-hard, implying that worst-case exponential complexity is unavoidable. We then identify two provably tractable settings: one based on sparsity in the interaction structure and another based on a stronger interaction-aware notion of sufficiency. Under either of these settings, we develop algorithms that compute provable explanations for neural networks with any-order interactions while retaining much of the computational efficiency of additive models. Empirically, we show that these explanations are significantly smaller and faster to obtain than those produced by standard algorithms, while being derived from models that achieve much higher accuracy than standard NAMs. Overall, our results help expand the class of neural networks that admit efficient provable explanations, while clarifying the fundamental role of feature interactions in shaping their complexity.

Authors: Idan Refaeli, Yizhak Yisrael Elboher, Guy Katz, Shahaf Bassan

Post-hoc explanation methods for neural networks are often heuristic and lack provable guarantees. A common alternative to such methods that was studied in recent years is to compute a cardinally-minimal subset of input features that is provably sufficient to determine the model's prediction. For general neural networks, however, finding such explanations is computationally intractable. In contrast, recent work showed that this problem becomes more tractable for a particular family of neural networks, namely neural additive models (NAMs), though standard NAMs are highly restrictive because they exclude feature interactions altogether. In this work, we study the computation of provably minimal and sufficient explanations for neural networks with restricted feature interactions, bridging the gap between fully general neural networks and purely additive models. We first prove that even with constant-order interactions, the problem is $Σ_2^P$-hard, implying that worst-case exponential complexity is unavoidable. We then identify two provably tractable settings: one based on sparsity in the interaction structure and another based on a stronger interaction-aware notion of sufficiency. Under either of these settings, we develop algorithms that compute provable explanations for neural networks with any-order interactions while retaining much of the computational efficiency of additive models. Empirically, we show that these explanations are significantly smaller and faster to obtain than those produced by standard algorithms, while being derived from models that achieve much higher accuracy than standard NAMs. Overall, our results help expand the class of neural networks that admit efficient provable explanations, while clarifying the fundamental role of feature interactions in shaping their complexity.

Tuesday, November 10 2026, 01:00

A Grover-Amplified Oracle for Maximum Clique on Graph, Minimum Hitting set and Dualizations on Hypergraphs Built on Quantum Phase Estimation

Authors: Francisco Piñero, Fernando L. \Pelayo, Hernan I. de la Cruz, Jose J. Paulet, Mauro Mezzini, Fernando Cuartero

In this paper, we use a quantum circuit called the Graph Encoding Operator which, applied to the complement $\bar G$ of any graph $G$, yields an oracle for the maximum clique problem suitable for amplification via Grover's algorithm. Analogously, we devise a Hypergraph Encoding Operator which, when applied to a hypergraph $\mathcal H$, yields an oracle for the minimum hitting set suitable for amplification via Grover's algorithm. Interestingly, the latter oracle provides, when measured, an approximation of the number of hitting sets of the hypergraph, thus solving the dualization problem for positive Boolean functions represented in a positive irredundant disjunctive normal form. We give the full construction of both operators, prove their correctness, analyze their complexity, and provide an implementation on a quantum simulator. We discuss the resulting complexity, contrast them with existing quantum computer algorithms based on Grover search and we will show that our algorithms outperform, to the best of our knowledge, all of the previous algorithms on space computational complexity by requiring only $O(N)$ number of qubits with respect to the number of vertices $N$, for representing both graphs and hypergraphs in the encoding operator and, consequently, results in an oracle for both graphs and hypergraphs that uses only $O(N)$ qubits even when the hypergraph contains an exponential (on the number of vertices) number of hyperedges.

Authors: Francisco Piñero, Fernando L. \Pelayo, Hernan I. de la Cruz, Jose J. Paulet, Mauro Mezzini, Fernando Cuartero

In this paper, we use a quantum circuit called the Graph Encoding Operator which, applied to the complement $\bar G$ of any graph $G$, yields an oracle for the maximum clique problem suitable for amplification via Grover's algorithm. Analogously, we devise a Hypergraph Encoding Operator which, when applied to a hypergraph $\mathcal H$, yields an oracle for the minimum hitting set suitable for amplification via Grover's algorithm. Interestingly, the latter oracle provides, when measured, an approximation of the number of hitting sets of the hypergraph, thus solving the dualization problem for positive Boolean functions represented in a positive irredundant disjunctive normal form. We give the full construction of both operators, prove their correctness, analyze their complexity, and provide an implementation on a quantum simulator. We discuss the resulting complexity, contrast them with existing quantum computer algorithms based on Grover search and we will show that our algorithms outperform, to the best of our knowledge, all of the previous algorithms on space computational complexity by requiring only $O(N)$ number of qubits with respect to the number of vertices $N$, for representing both graphs and hypergraphs in the encoding operator and, consequently, results in an oracle for both graphs and hypergraphs that uses only $O(N)$ qubits even when the hypergraph contains an exponential (on the number of vertices) number of hyperedges.

Tuesday, November 10 2026, 01:00

Initialization masks overcome decoding obstructions in compressed QAOA

Authors: Emilio Semre, Steven Frankel

In compressed quantum optimization, a variational state is converted into a classical assignment by decoding noncommuting observables. We show that this step creates structural restrictions that changing the initialization can remove, even when increasing circuit depth cannot. For an explicit infinite family of connected cubic graphs on $n=3r$ vertices, with a fixed Pauli encoding and deterministic sign decoder, every all-plus QAOA circuit has cut at most $2r$, at every finite depth and every real angle sequence. A uniform signed product-state mask at fixed depth-one angles has expected cut $13r/4$, an additive advantage $5n/12$; an infinite subfamily admits an optimal mask. A second construction retains a linear depth-one separation under weak weighted couplings, a bounded cost-angle domain and a specified production-encoder ordering. Noiseless simulations then test classical mask selection before energy optimization. On 1,699 random cubic MaxCut instances, one selected mask achieves mean ratio $0.9100$, versus $0.7067$ for the all-plus $X$ baseline and $0.9125$ for the best of $r^2$ separately optimized random masks. On complete cohorts of weighted MaxCut, sparse Ising, Max-2-XOR and Max-2-SAT, the selected mask also improves on the baseline at every size, but it trails the pool at small sizes and leads it significantly only at the larger MaxCut and Ising sizes. Finite-pool order statistics separate the effect of selection from that of drawing a random mask. The results distinguish decoded expressivity from optimized energy and empirical performance; they do not establish a quantum computational advantage.

Authors: Emilio Semre, Steven Frankel

In compressed quantum optimization, a variational state is converted into a classical assignment by decoding noncommuting observables. We show that this step creates structural restrictions that changing the initialization can remove, even when increasing circuit depth cannot. For an explicit infinite family of connected cubic graphs on $n=3r$ vertices, with a fixed Pauli encoding and deterministic sign decoder, every all-plus QAOA circuit has cut at most $2r$, at every finite depth and every real angle sequence. A uniform signed product-state mask at fixed depth-one angles has expected cut $13r/4$, an additive advantage $5n/12$; an infinite subfamily admits an optimal mask. A second construction retains a linear depth-one separation under weak weighted couplings, a bounded cost-angle domain and a specified production-encoder ordering. Noiseless simulations then test classical mask selection before energy optimization. On 1,699 random cubic MaxCut instances, one selected mask achieves mean ratio $0.9100$, versus $0.7067$ for the all-plus $X$ baseline and $0.9125$ for the best of $r^2$ separately optimized random masks. On complete cohorts of weighted MaxCut, sparse Ising, Max-2-XOR and Max-2-SAT, the selected mask also improves on the baseline at every size, but it trails the pool at small sizes and leads it significantly only at the larger MaxCut and Ising sizes. Finite-pool order statistics separate the effect of selection from that of drawing a random mask. The results distinguish decoded expressivity from optimized energy and empirical performance; they do not establish a quantum computational advantage.

Tuesday, November 10 2026, 01:00

HexIt: Learning Hexahedral Block Decomposition with Expert Iteration on Combinatorial Maps

Authors: Arjun Narayanan, Per-Olof Persson

Decomposing a polyhedral solid into a few well-shaped hexahedral blocks is a long-standing open problem in mesh generation. We cast it as a sequential decision problem on a 3-dimensional generalised map: an agent splits edges, faces and volumes with three insert operations until every cell is a hexahedron and the mesh passes a quality threshold, and a policy that convolves over the map's involutions chooses each insert. A successful decomposition takes over a hundred decisions, and a uniformly random policy never finds one. We clone a policy from constructed decompositions and improve it by expert iteration. On held-out chamfered solids that share no boundary topology with the cloning data, the final policy solves an expected $94.4\%$ in $20$ samples, and expert iteration raises the best-of-20 solve rate on large chamfered solids from $0.203$ to $0.355$. At matched compute, PPO, GRPO and VinePPO updates all make the cloned policy worse.

Authors: Arjun Narayanan, Per-Olof Persson

Decomposing a polyhedral solid into a few well-shaped hexahedral blocks is a long-standing open problem in mesh generation. We cast it as a sequential decision problem on a 3-dimensional generalised map: an agent splits edges, faces and volumes with three insert operations until every cell is a hexahedron and the mesh passes a quality threshold, and a policy that convolves over the map's involutions chooses each insert. A successful decomposition takes over a hundred decisions, and a uniformly random policy never finds one. We clone a policy from constructed decompositions and improve it by expert iteration. On held-out chamfered solids that share no boundary topology with the cloning data, the final policy solves an expected $94.4\%$ in $20$ samples, and expert iteration raises the best-of-20 solve rate on large chamfered solids from $0.203$ to $0.355$. At matched compute, PPO, GRPO and VinePPO updates all make the cloned policy worse.

Tuesday, November 10 2026, 01:00

Sets of reach $R$ are $\operatorname{CAT}\left( 1/R^2\right)$

Authors: André Lieutier, Mathijs Wintraecken

Positive reach underpins many results in computational geometry and topology. It is used for triangulation criteria, topological inference, and manifold learning. The geometric properties of these sets have therefore been studied intensely. In this paper we focus on the intrinsic metric properties of sets of positive reach. In particular we give an elementary proof of the fact that (closed) sets of reach at least $R$ are $\operatorname{CAT}(1/R^2)$. This fact was established earlier by Alexander and Bishop, using quite significant machinery. The fact that a space is $\operatorname{CAT}(1/R^2)$ offers many advantages, such as the uniqueness and continuity of sufficiently short geodesics, the existence of Fr{é}chet means and by extension the Rips based homotopy reconstruction of Hausmann is available.

Authors: André Lieutier, Mathijs Wintraecken

Positive reach underpins many results in computational geometry and topology. It is used for triangulation criteria, topological inference, and manifold learning. The geometric properties of these sets have therefore been studied intensely. In this paper we focus on the intrinsic metric properties of sets of positive reach. In particular we give an elementary proof of the fact that (closed) sets of reach at least $R$ are $\operatorname{CAT}(1/R^2)$. This fact was established earlier by Alexander and Bishop, using quite significant machinery. The fact that a space is $\operatorname{CAT}(1/R^2)$ offers many advantages, such as the uniqueness and continuity of sufficiently short geodesics, the existence of Fr{é}chet means and by extension the Rips based homotopy reconstruction of Hausmann is available.

Tuesday, November 10 2026, 01:00

Lower Bounds Above $\sqrt{3}$ for Online Makespan Minimization on Four Identical Machines

Authors: Maximilian Janke

For deterministic online makespan minimization, $m = 4$ identical machines is the smallest case whose optimal competitive ratio is open: it is known only to lie between $\sqrt{3} \approx 1.73205$ and $26/15 \approx 1.73333$. We prove a lower bound of $14383/8304 \approx 1.73206$ using a single branching layer, and refine the construction with a second, mirrored branching layer to obtain $14350/8283 \approx 1.73246$, closing about a third of the gap. Our constructions are the first branching adversaries for $m = 4$ machines, and the first explicit lower bounds for online makespan minimization in which the adversary presents jobs in non-monotone order of size.

Authors: Maximilian Janke

For deterministic online makespan minimization, $m = 4$ identical machines is the smallest case whose optimal competitive ratio is open: it is known only to lie between $\sqrt{3} \approx 1.73205$ and $26/15 \approx 1.73333$. We prove a lower bound of $14383/8304 \approx 1.73206$ using a single branching layer, and refine the construction with a second, mirrored branching layer to obtain $14350/8283 \approx 1.73246$, closing about a third of the gap. Our constructions are the first branching adversaries for $m = 4$ machines, and the first explicit lower bounds for online makespan minimization in which the adversary presents jobs in non-monotone order of size.

Tuesday, November 10 2026, 01:00

Sparse Predict and Demote: Optimal Scheduling with Minimal Size Advice

Authors: Chengyi Liu, Thodoris Lykouris, Junxuan Shen, Adam Wierman

Knowledge of job sizes is crucial for optimizing the mean response time in single-server scheduling. However, predictions of job sizes are typically inexact and may be costly to obtain. Thus, a long-standing research question is: what is the minimal information about job sizes required to obtain near-optimal mean response time? We introduce a novel policy, \emph{Sparse Predict and Demote (SPD)}, and prove that it achieves order-optimal mean response time in heavy traffic while requiring only one-bit job-size predictions for a vanishing fraction of arrivals. SPD achieves heavy-traffic order-optimality even in situations when predictions are (i) inexact up to any fixed multiplicative error bound, and (ii) costly, i.e., the server is used to compute predictions and is thus unavailable to serve jobs while processing prediction queries. Finally, we use numerical experiments to illustrate that SPD maintains near-optimal performance beyond the heavy-traffic regime.

Authors: Chengyi Liu, Thodoris Lykouris, Junxuan Shen, Adam Wierman

Knowledge of job sizes is crucial for optimizing the mean response time in single-server scheduling. However, predictions of job sizes are typically inexact and may be costly to obtain. Thus, a long-standing research question is: what is the minimal information about job sizes required to obtain near-optimal mean response time? We introduce a novel policy, \emph{Sparse Predict and Demote (SPD)}, and prove that it achieves order-optimal mean response time in heavy traffic while requiring only one-bit job-size predictions for a vanishing fraction of arrivals. SPD achieves heavy-traffic order-optimality even in situations when predictions are (i) inexact up to any fixed multiplicative error bound, and (ii) costly, i.e., the server is used to compute predictions and is thus unavailable to serve jobs while processing prediction queries. Finally, we use numerical experiments to illustrate that SPD maintains near-optimal performance beyond the heavy-traffic regime.

Tuesday, November 10 2026, 01:00

Regev's Reduction Reaches Reconstruction

Authors: Stephen P. Jordan, Quynh T. Nguyen, Leo Zhou

Regev's reduction uses a quantum Fourier transform to carry out reductions between various problems related to lattices and codes. We apply it to the problem of sampling from Gibbs distributions of certain classical spin glass Hamiltonians, which can equivalently be thought of as cost functions for average-case instances of an optimization problem called $D$-regular max-$k$-LINSAT. This problem becomes hard at low temperature. Here, we use a quantum decoding algorithm called Belief Propagation with Quantum Messages (BPQM) to solve the decoding problem arising from Regev's reduction. Our main contribution is a proof that the minimum temperature reachable using Regev+BPQM is exactly equal to the reconstruction threshold temperature for noisy channels transmitting signals on a tree network. This equality can be understood via Renes' duality between classical and quantum channels, which we use to show that quantum decoding is possible exactly when classical reconstruction on the dual problem is impossible. M{é}zard and Montanari identified this reconstruction temperature with a dynamical glass transition predicted by the one-step replica-symmetry breaking (1RSB) cavity equation. It is conjectured that local samplers such as Metropolis Monte Carlo and Glauber dynamics cannot equilibrate efficiently below this reconstruction temperature but may do so above it; our large numerical experiments show close agreement with this prediction across hundreds of different problem parameters. Therefore, our results rigorously show that Regev+BPQM reaches a temperature for which, to our knowledge, there is no matching classical guarantee, and brings Regev+BPQM to the conjectured limit of efficient Gibbs sampling by classical local dynamics.

Authors: Stephen P. Jordan, Quynh T. Nguyen, Leo Zhou

Regev's reduction uses a quantum Fourier transform to carry out reductions between various problems related to lattices and codes. We apply it to the problem of sampling from Gibbs distributions of certain classical spin glass Hamiltonians, which can equivalently be thought of as cost functions for average-case instances of an optimization problem called $D$-regular max-$k$-LINSAT. This problem becomes hard at low temperature. Here, we use a quantum decoding algorithm called Belief Propagation with Quantum Messages (BPQM) to solve the decoding problem arising from Regev's reduction. Our main contribution is a proof that the minimum temperature reachable using Regev+BPQM is exactly equal to the reconstruction threshold temperature for noisy channels transmitting signals on a tree network. This equality can be understood via Renes' duality between classical and quantum channels, which we use to show that quantum decoding is possible exactly when classical reconstruction on the dual problem is impossible. M{é}zard and Montanari identified this reconstruction temperature with a dynamical glass transition predicted by the one-step replica-symmetry breaking (1RSB) cavity equation. It is conjectured that local samplers such as Metropolis Monte Carlo and Glauber dynamics cannot equilibrate efficiently below this reconstruction temperature but may do so above it; our large numerical experiments show close agreement with this prediction across hundreds of different problem parameters. Therefore, our results rigorously show that Regev+BPQM reaches a temperature for which, to our knowledge, there is no matching classical guarantee, and brings Regev+BPQM to the conjectured limit of efficient Gibbs sampling by classical local dynamics.

Tuesday, November 10 2026, 01:00

Randomized Greedy MIS Is Fast at Every Node of a Tree

Authors: Mina Dalirrooyfard, Konstantin Makarychev, Slobodan Mitrović, Srikkanth Ramachandran

The Randomized Greedy MIS (RGMIS) algorithm computes a maximal independent set (MIS) by processing the vertices in a uniformly random order and greedily adding each vertex to the independent set whenever possible. Despite its simplicity, this algorithm has proved effective in a range of parallel, distributed, and dynamic settings. For sufficiently small maximum degree, Nguyen and Onak (FOCS~'08) and Yoshida, Yamamoto, and Ito (STOC~'09) showed that a recursive simulation of RGMIS can be used to estimate the sizes of matching and vertex cover in sublinear time. The key subroutine in their approach is the following local query procedure: given a vertex $v$, to determine whether $v$ belongs to the MIS, recursively evaluate the neighbors of $v$ that precede $v$ in the random order. If any such neighbor is found to belong to the MIS, the procedure immediately concludes that $v$ is not in the MIS; otherwise, it concludes that $v$ is in the MIS. For graphs of maximum degree $Δ$, Nguyen and Onak showed that, in expectation over the random ordering, this procedure makes $O(\exp(Δ))$ recursive calls from a worst-case starting vertex. No better upper bound is known even for trees. We prove that on trees, from \textit{any} starting vertex, this recursive RGMIS procedure makes $\mathrm{poly}(Δ)$ recursive calls in expectation.

Authors: Mina Dalirrooyfard, Konstantin Makarychev, Slobodan Mitrović, Srikkanth Ramachandran

The Randomized Greedy MIS (RGMIS) algorithm computes a maximal independent set (MIS) by processing the vertices in a uniformly random order and greedily adding each vertex to the independent set whenever possible. Despite its simplicity, this algorithm has proved effective in a range of parallel, distributed, and dynamic settings. For sufficiently small maximum degree, Nguyen and Onak (FOCS~'08) and Yoshida, Yamamoto, and Ito (STOC~'09) showed that a recursive simulation of RGMIS can be used to estimate the sizes of matching and vertex cover in sublinear time. The key subroutine in their approach is the following local query procedure: given a vertex $v$, to determine whether $v$ belongs to the MIS, recursively evaluate the neighbors of $v$ that precede $v$ in the random order. If any such neighbor is found to belong to the MIS, the procedure immediately concludes that $v$ is not in the MIS; otherwise, it concludes that $v$ is in the MIS. For graphs of maximum degree $Δ$, Nguyen and Onak showed that, in expectation over the random ordering, this procedure makes $O(\exp(Δ))$ recursive calls from a worst-case starting vertex. No better upper bound is known even for trees. We prove that on trees, from \textit{any} starting vertex, this recursive RGMIS procedure makes $\mathrm{poly}(Δ)$ recursive calls in expectation.

Tuesday, November 10 2026, 01:00

Strong Connectivity Oracle for Multiple Failures via Rooted Flow-Augmentation

Authors: Aditya Anand, Thatchaphol Saranurak, Benyu Wang

We present a significantly improved $k$-fault-tolerant strong connectivity oracle, one of the most basic data structures on a directed graph $G$ that, given any set $F$ of failed edges of size $|F|\le k$, answers whether $G \setminus F$ is strongly connected. For every constant $k$, our oracle only uses $n^{1+o(1)}$ space and $n^{o(1)}$ query time. No prior result can achieve $o(n^{2})$ space and $o(n)$ query time, \emph{even when $k=2$}. Previous oracles either require $Ω(n)$ query time [BCR18], or $Ω(n^2)$ space [vdBS19]. Our main technical contribution is a \emph{rooted} version of the highly successful flow augmentation technique, which may be of independent interest. Given a directed graph $G$, an integer $k$, and a source vertex $s$, we can sample a set of edges $A$ such that each minimal $s$-rooted cut of size at most $k$ becomes an $s$-rooted min-cut in $G + A$ with probability $2^{-poly(k)}n^{-o(1)}$.

Authors: Aditya Anand, Thatchaphol Saranurak, Benyu Wang

We present a significantly improved $k$-fault-tolerant strong connectivity oracle, one of the most basic data structures on a directed graph $G$ that, given any set $F$ of failed edges of size $|F|\le k$, answers whether $G \setminus F$ is strongly connected. For every constant $k$, our oracle only uses $n^{1+o(1)}$ space and $n^{o(1)}$ query time. No prior result can achieve $o(n^{2})$ space and $o(n)$ query time, \emph{even when $k=2$}. Previous oracles either require $Ω(n)$ query time [BCR18], or $Ω(n^2)$ space [vdBS19]. Our main technical contribution is a \emph{rooted} version of the highly successful flow augmentation technique, which may be of independent interest. Given a directed graph $G$, an integer $k$, and a source vertex $s$, we can sample a set of edges $A$ such that each minimal $s$-rooted cut of size at most $k$ becomes an $s$-rooted min-cut in $G + A$ with probability $2^{-poly(k)}n^{-o(1)}$.

Tuesday, November 10 2026, 01:00

Flexible Edge Cover: A Disjunctive Programming driven Approximation

Authors: Karthekeyan Chandrasekaran, Krishna Kalathur

We introduce the flexible edge cover problem, denoted FLEX-EDGE-COVER: the input here is an undirected graph with non-negative edge costs along with a partition of the edges into safe and unsafe edges, non-negative integers $p$ and $q$, and the goal is to find a minimum cost subset of edges such that each vertex has either at least $p$ safe edges incident to it or at least $p+q$ total edges incident to it. We show that FLEX-EDGE-COVER is APX-hard in simple bipartite graphs. A greedy approach leads to a $2$-approximation in all graphs. We show that the $2$-factor can be improved in simple graphs via a novel disjunctive programming-based LP formulation. AI-tools were used in the analysis of our rounding algorithm. Our motivation for introducing and investigating FLEX-EDGE-COVER is its close connection to the Flexible Graph Connectivity problem (denoted $(p,q)$-FGC) which has garnered much recent attention. For $(p,q)$-FGC, we develop a tighter LP-formulation via the disjunctive programming approach. We hope that our disjunctive-programming based LP and rounding algorithms for FLEX-EDGE-COVER will serve as a starting point towards a constant factor approximation for $(p,q)$-FGC.

Authors: Karthekeyan Chandrasekaran, Krishna Kalathur

We introduce the flexible edge cover problem, denoted FLEX-EDGE-COVER: the input here is an undirected graph with non-negative edge costs along with a partition of the edges into safe and unsafe edges, non-negative integers $p$ and $q$, and the goal is to find a minimum cost subset of edges such that each vertex has either at least $p$ safe edges incident to it or at least $p+q$ total edges incident to it. We show that FLEX-EDGE-COVER is APX-hard in simple bipartite graphs. A greedy approach leads to a $2$-approximation in all graphs. We show that the $2$-factor can be improved in simple graphs via a novel disjunctive programming-based LP formulation. AI-tools were used in the analysis of our rounding algorithm. Our motivation for introducing and investigating FLEX-EDGE-COVER is its close connection to the Flexible Graph Connectivity problem (denoted $(p,q)$-FGC) which has garnered much recent attention. For $(p,q)$-FGC, we develop a tighter LP-formulation via the disjunctive programming approach. We hope that our disjunctive-programming based LP and rounding algorithms for FLEX-EDGE-COVER will serve as a starting point towards a constant factor approximation for $(p,q)$-FGC.

Tuesday, November 10 2026, 01:00

Fully Dynamic Correlation Clustering with Vertex Updates in Polylogarithmic Update Time

Authors: Vladimir Braverman, Prathamesh Dharangutte, Chenglin Fan, Chen Wang, Shenghao Xie, Samson Zhou

We present a fully dynamic algorithm for correlation clustering that supports \emph{vertex} insertions and deletions while maintaining an $O(1)$ approximation in polylogarithmic amortized update time. This resolves an open problem of vertex deletions posed by a recent line of work on dynamic correlation clustering. We adopt the min-disagreement objective for correlation clustering, i.e., given a signed complete graph $G=(V,E^+\cup E^-)$, we want to minimize the total number of $(+)$ edge crossing clusters and $(-)$ edges within clusters. The input is given in the adjacency list model, where each update inserts or deletes a node along with its full set of $(+)$ neighbors. Our main technical contribution is a fault-tolerant sparse-dense decomposition framework which unlike the existing decomposition-based frameworks allows for structural inconsistencies as long as the induced cost is bounded. We also show a separation between explicitly maintaining the (stronger) sparse-dense decomposition and maintaining a constant-factor approximation to correlation clustering, which may be of independent interest.

Authors: Vladimir Braverman, Prathamesh Dharangutte, Chenglin Fan, Chen Wang, Shenghao Xie, Samson Zhou

We present a fully dynamic algorithm for correlation clustering that supports \emph{vertex} insertions and deletions while maintaining an $O(1)$ approximation in polylogarithmic amortized update time. This resolves an open problem of vertex deletions posed by a recent line of work on dynamic correlation clustering. We adopt the min-disagreement objective for correlation clustering, i.e., given a signed complete graph $G=(V,E^+\cup E^-)$, we want to minimize the total number of $(+)$ edge crossing clusters and $(-)$ edges within clusters. The input is given in the adjacency list model, where each update inserts or deletes a node along with its full set of $(+)$ neighbors. Our main technical contribution is a fault-tolerant sparse-dense decomposition framework which unlike the existing decomposition-based frameworks allows for structural inconsistencies as long as the induced cost is bounded. We also show a separation between explicitly maintaining the (stronger) sparse-dense decomposition and maintaining a constant-factor approximation to correlation clustering, which may be of independent interest.

Tuesday, November 10 2026, 01:00

A Note on the Hardness of Partially Dynamic Max Flow

Authors: Thatchaphol Saranurak, Benyu Wang

We study the fine-grained complexity of partially dynamic max flow in capacitated graphs. Assuming the combinatorial 4-clique hypothesis, combinatorial algorithms require $Ω(n^{2-o(1)})$ amortized update time. Thus, recomputing max flow after each update is optimal up to subpolynomial factors for combinatorial algorithms in dense graphs. The previous conditional lower bound was $Ω(n^{1-o(1)})$ amortized update time under SETH or the OMv hypothesis and applies to all algorithms [Dah16].

Authors: Thatchaphol Saranurak, Benyu Wang

We study the fine-grained complexity of partially dynamic max flow in capacitated graphs. Assuming the combinatorial 4-clique hypothesis, combinatorial algorithms require $Ω(n^{2-o(1)})$ amortized update time. Thus, recomputing max flow after each update is optimal up to subpolynomial factors for combinatorial algorithms in dense graphs. The previous conditional lower bound was $Ω(n^{1-o(1)})$ amortized update time under SETH or the OMv hypothesis and applies to all algorithms [Dah16].

Tuesday, November 10 2026, 01:00

Dynamic Edge Orientation on Low-Arboricity Graphs: Breaking the $\log n$ Barrier

Authors: Sam Hiken, Nicole Wein

We study the dynamic edge orientation problem introduced in a seminal 1999 paper by Brodal and Fagerberg: given a fully dynamic graph $G$ with arboricity bounded by $α$, maintain an orientation on $G$'s edges so every vertex has out-degree upper bounded by a function of $α$. They provided an algorithm with $O(α+ \log n)$ amortized update time that maintains an orientation with maximum out-degree $O(α)$. In the decades since, this algorithm has found widespread applications in dynamic graph algorithms. Nevertheless, it has remained a stubborn open question whether, when $α= o(\log n)$, the $O(\log n)$ upper bound can be improved. We answer this 27-year-old question in the affirmative, showing that Brodal and Fagerberg's algorithm for maintaining an out-degree $O(α)$ orientation runs in $O(α\cdot \log \log n)$ amortized update time, which is an exponentially-improved dependence on $n$. This is the first improvement on Brodal and Fagerberg's upper bound even for the $α=1$ case of dynamic forests. Our crucial technical innovation is to construct a hierarchy of dynamic graphs with a property we call near isolation. We believe that this construction may be useful for related problems.

Authors: Sam Hiken, Nicole Wein

We study the dynamic edge orientation problem introduced in a seminal 1999 paper by Brodal and Fagerberg: given a fully dynamic graph $G$ with arboricity bounded by $α$, maintain an orientation on $G$'s edges so every vertex has out-degree upper bounded by a function of $α$. They provided an algorithm with $O(α+ \log n)$ amortized update time that maintains an orientation with maximum out-degree $O(α)$. In the decades since, this algorithm has found widespread applications in dynamic graph algorithms. Nevertheless, it has remained a stubborn open question whether, when $α= o(\log n)$, the $O(\log n)$ upper bound can be improved. We answer this 27-year-old question in the affirmative, showing that Brodal and Fagerberg's algorithm for maintaining an out-degree $O(α)$ orientation runs in $O(α\cdot \log \log n)$ amortized update time, which is an exponentially-improved dependence on $n$. This is the first improvement on Brodal and Fagerberg's upper bound even for the $α=1$ case of dynamic forests. Our crucial technical innovation is to construct a hierarchy of dynamic graphs with a property we call near isolation. We believe that this construction may be useful for related problems.

Tuesday, November 10 2026, 01:00

Polynomial-Space Quantum Algorithms from Classical Branching: Colouring, Domination and Monotone Local Search

Authors: Sebastian Senge

We give quantum algorithms for exponential-time problems on $n$ vertices or elements, with bounded error, polynomial space and no quantum random access memory. Our results are these. (1) 4-Colouring in quantum time $O(1.3032^n)$, from a polynomial-space branching algorithm with polynomial-space terminal algorithms; with the branching algorithm alone, $O(1.3100^n)$. The previous best bound in this model, to our knowledge, is $O(1.3231^n)$. (2) $k$-Colouring below the previous best bound for $k \in \{5, 7, 8, 9, 10, 11\}$ and $13 \le k \le 18$, from $O(1.4518^n)$ for $k = 5$ to $O(1.9233^n)$ for $k = 18$. (3) Dominating Set in $O(1.2235^n)$, against the previous $O(2^{n/2})$, Independent Dominating Set in $O(1.1583^n)$, and Edge Dominating Set and Minimum Maximal Matching in $O(1.1501^n)$. (4) A quantum form of monotone local search: if the extension problem has a backtracking tree with $c^k$ nodes up to polynomial factors, the problem is solved in quantum time $(2 - 1/c)^{n/2}$ up to polynomial factors. This gives Feedback Vertex Set in $O(1.3081^n)$, Subset and Group Feedback Vertex Set in $O(1.3229^n)$, Cluster Vertex Deletion in $O(1.2033^n)$, 3-Hitting Set in $O(1.2405^n)$, Proper Interval Vertex Deletion in $O(1.3541^n)$, Multicut on Trees in $O(1.1647^n)$ and Min-Ones $d$-SAT in $O^*((2-1/d)^{n/2})$. (5) Quantum exponential-time approximations below the quantum exact bounds, such as a 1.1-approximation for Vertex Cover in $O(1.0555^n)$ and for Feedback Vertex Set in $O(1.2161^n)$. The colouring exponents for $k \ge 6$ come from an evaluation of a known reduction recursion, computed in floating point with a margin.

Authors: Sebastian Senge

We give quantum algorithms for exponential-time problems on $n$ vertices or elements, with bounded error, polynomial space and no quantum random access memory. Our results are these. (1) 4-Colouring in quantum time $O(1.3032^n)$, from a polynomial-space branching algorithm with polynomial-space terminal algorithms; with the branching algorithm alone, $O(1.3100^n)$. The previous best bound in this model, to our knowledge, is $O(1.3231^n)$. (2) $k$-Colouring below the previous best bound for $k \in \{5, 7, 8, 9, 10, 11\}$ and $13 \le k \le 18$, from $O(1.4518^n)$ for $k = 5$ to $O(1.9233^n)$ for $k = 18$. (3) Dominating Set in $O(1.2235^n)$, against the previous $O(2^{n/2})$, Independent Dominating Set in $O(1.1583^n)$, and Edge Dominating Set and Minimum Maximal Matching in $O(1.1501^n)$. (4) A quantum form of monotone local search: if the extension problem has a backtracking tree with $c^k$ nodes up to polynomial factors, the problem is solved in quantum time $(2 - 1/c)^{n/2}$ up to polynomial factors. This gives Feedback Vertex Set in $O(1.3081^n)$, Subset and Group Feedback Vertex Set in $O(1.3229^n)$, Cluster Vertex Deletion in $O(1.2033^n)$, 3-Hitting Set in $O(1.2405^n)$, Proper Interval Vertex Deletion in $O(1.3541^n)$, Multicut on Trees in $O(1.1647^n)$ and Min-Ones $d$-SAT in $O^*((2-1/d)^{n/2})$. (5) Quantum exponential-time approximations below the quantum exact bounds, such as a 1.1-approximation for Vertex Cover in $O(1.0555^n)$ and for Feedback Vertex Set in $O(1.2161^n)$. The colouring exponents for $k \ge 6$ come from an evaluation of a known reduction recursion, computed in floating point with a margin.

Tuesday, November 10 2026, 01:00

Strong Backdoor to Forests is FPT: Tight Linear-Time FPT for Essentially General SAT

Authors: Olesia Bilyk, Aidar Munduzbaev, Roohani Sharma, Mamnoon Siam

In a seminal work, Gaspers and Szeider [FOCS 2013] gave a fixed-parameter tractable (FPT) approximate algorithm for finding a strong backdoor of size at most $k$ to incidence treewidth at most $t$, for a given CNF-formula $φ$. In particular, their algorithm runs in $f(k,t) |φ|^3$ time, and finds a strong backdoor to treewidth at most $t$, of $φ$, of size at most $2^k$ whenever there exists one of size at most $k$. Determining whether the detection of strong backdoors to bounded incidence treewidth is FPT parameterized by $k+t$ was left open in their work and this has been repeatedly posed as an open question by Gaspers and Szeider, for example in [The Multivariate Algorithmic Revolution and Beyond 2012] and more recently in [Computer Science Reviews 2025], even for incidence forests and when the input is a $3$-CNF formula. We answer this question affirmatively. We show that strong backdoors to forests can be found in linear-FPT time on arbitrary non-tautological CNF formulas (not just on 3-CNF formulas). Given such a formula $φ$ and a budget $k$, our algorithm finds a strong backdoor to the class of forests, of size at most $k$, or correctly reports that none exists, in $2^{2^{O(k)}} |φ|$ time. Moreover, we show that our result is as tight as it can get. First, we show that finding strong backdoors to treewidth $t$ for any $t \geq 2$ is W[1]-hard, even on non-tautological 3-CNF formulas. Second, the running time of our algorithm is tight: it is linear in the input size and a better than double-exponential dependence on $k$ in the running time would imply a $2^{o(N/\log N)}$-time algorithm for solving 3-SAT on $N$ variables, which would beat even the very recently claimed sub-exponential algorithm for solving 3-SAT by OpenAI and Anthropic.

Authors: Olesia Bilyk, Aidar Munduzbaev, Roohani Sharma, Mamnoon Siam

In a seminal work, Gaspers and Szeider [FOCS 2013] gave a fixed-parameter tractable (FPT) approximate algorithm for finding a strong backdoor of size at most $k$ to incidence treewidth at most $t$, for a given CNF-formula $φ$. In particular, their algorithm runs in $f(k,t) |φ|^3$ time, and finds a strong backdoor to treewidth at most $t$, of $φ$, of size at most $2^k$ whenever there exists one of size at most $k$. Determining whether the detection of strong backdoors to bounded incidence treewidth is FPT parameterized by $k+t$ was left open in their work and this has been repeatedly posed as an open question by Gaspers and Szeider, for example in [The Multivariate Algorithmic Revolution and Beyond 2012] and more recently in [Computer Science Reviews 2025], even for incidence forests and when the input is a $3$-CNF formula. We answer this question affirmatively. We show that strong backdoors to forests can be found in linear-FPT time on arbitrary non-tautological CNF formulas (not just on 3-CNF formulas). Given such a formula $φ$ and a budget $k$, our algorithm finds a strong backdoor to the class of forests, of size at most $k$, or correctly reports that none exists, in $2^{2^{O(k)}} |φ|$ time. Moreover, we show that our result is as tight as it can get. First, we show that finding strong backdoors to treewidth $t$ for any $t \geq 2$ is W[1]-hard, even on non-tautological 3-CNF formulas. Second, the running time of our algorithm is tight: it is linear in the input size and a better than double-exponential dependence on $k$ in the running time would imply a $2^{o(N/\log N)}$-time algorithm for solving 3-SAT on $N$ variables, which would beat even the very recently claimed sub-exponential algorithm for solving 3-SAT by OpenAI and Anthropic.

Tuesday, November 10 2026, 01:00

Amortized Cell-Probe Lower Bounds for Dynamic Boolean Problems

Authors: Young Kun Ko

We prove the first unconditional amortized $Ω\big((\log N/\log\log N)^2\big)$ cell-probe lower bound for a Boolean dynamic problem. For insert-only multiphase Disjointness with $N$ query sets over $[N]$ and word size $w=Θ(\log N)$, a fixed distribution of interleaved insertions and queries forces this expected average cost per operation, excluding preprocessing. The bound allows sufficiently small constant error for each query of every fixed legal sequence and imposes no restriction on individual operation costs. It transfers to incremental directed reachability, incremental subgraph connectivity, and fully dynamic triangle detection. We also prove $t_q=Ω\big((\log N/\log(t_uw))^2\big)$ for amortized insertion time $1\le t_u=N^{o(1)}$ and expected query time $t_q$, when the number of insertions before the final query is not announced in advance. We reach the quadratic-logarithmic scale with bounded error by combining interleaving with the 2.5-round communication framework [Ko26a, Ko26b] and controlling errors among accepted simulations.

Authors: Young Kun Ko

We prove the first unconditional amortized $Ω\big((\log N/\log\log N)^2\big)$ cell-probe lower bound for a Boolean dynamic problem. For insert-only multiphase Disjointness with $N$ query sets over $[N]$ and word size $w=Θ(\log N)$, a fixed distribution of interleaved insertions and queries forces this expected average cost per operation, excluding preprocessing. The bound allows sufficiently small constant error for each query of every fixed legal sequence and imposes no restriction on individual operation costs. It transfers to incremental directed reachability, incremental subgraph connectivity, and fully dynamic triangle detection. We also prove $t_q=Ω\big((\log N/\log(t_uw))^2\big)$ for amortized insertion time $1\le t_u=N^{o(1)}$ and expected query time $t_q$, when the number of insertions before the final query is not announced in advance. We reach the quadratic-logarithmic scale with bounded error by combining interleaving with the 2.5-round communication framework [Ko26a, Ko26b] and controlling errors among accepted simulations.

Tuesday, November 10 2026, 01:00

Solving Large Rubik's Cubes within a Factor of $2.37+o(1)$ of Optimal

Authors: Jeremy Mrzyglocki, Tomas Rokicki, Maxine Perroni-Scharf, Erik D. Demaine

We design optimized algorithms to solve the $n \times n \times n$ Rubik's Cube within $30.5 n^2 / \lg n + o(n^2 / \lg n)$ moves. This upper bound is asymptotically within a factor of $2.37$ of the known lower bound, which applies to almost all instances. The constant $30.5$ is also the first explicit constant factor for the known upper bound of $O(n^2 / \lg n)$, where the implicit constant of the previous algorithm was $1.01 \cdot 10^{40}$ (ten duodecillion). Thus we improve the upper-bound constant by a factor of more than $3.30 \cdot 10^{38}$ (330 undecillion). Our algorithm runs in time linear in the input size $N = Θ(n^2)$, and uses $O(\sqrt{N} \lg N)$ extra space (beyond the state of the cube). Our implementation is the first practical cube solver to generate solutions of length $O(n^2/\lg n)$. Our latest solver outperforms existing solvers for $n$ as small as $840$.

Authors: Jeremy Mrzyglocki, Tomas Rokicki, Maxine Perroni-Scharf, Erik D. Demaine

We design optimized algorithms to solve the $n \times n \times n$ Rubik's Cube within $30.5 n^2 / \lg n + o(n^2 / \lg n)$ moves. This upper bound is asymptotically within a factor of $2.37$ of the known lower bound, which applies to almost all instances. The constant $30.5$ is also the first explicit constant factor for the known upper bound of $O(n^2 / \lg n)$, where the implicit constant of the previous algorithm was $1.01 \cdot 10^{40}$ (ten duodecillion). Thus we improve the upper-bound constant by a factor of more than $3.30 \cdot 10^{38}$ (330 undecillion). Our algorithm runs in time linear in the input size $N = Θ(n^2)$, and uses $O(\sqrt{N} \lg N)$ extra space (beyond the state of the cube). Our implementation is the first practical cube solver to generate solutions of length $O(n^2/\lg n)$. Our latest solver outperforms existing solvers for $n$ as small as $840$.

Tuesday, November 10 2026, 01:00

Optimal Injectivity of the SparseStack

Authors: Lorenzo Beretta, Simone Bombari, Margalit Glasgow, Raphael Arkady Meyer, Kijun Shin, David P. Woodruff

An Oblivious Subspace Injection (OSI) is a random sketching matrix that reliably preserves the minimum singular value of any fixed subspace with high constant probability, requiring only a one-sided spectral lower bound without controlling upper singular values. While classical Oblivious Subspace Embeddings (OSEs) fundamentally require row sparsity $Ω(\log r)$ at linear sketch dimension $k = O(r)$, we show that one-sided injectivity permits strictly sparser sketches, establishing matching (up to $\log\log r$ factors) upper and lower bounds of $\tildeΘ(\sqrt{\log r})$. First, we prove that the SparseStack sketch achieves constant injectivity at linear dimension $k = Θ(r)$ with row sparsity only $ζ= O(\sqrt{\log r})$, well below the $Θ(\log r)$ barrier where standard operator-norm concentration fails. Second, we establish a matching lower bound: any isotropic random sketching matrix satisfying a mild row-overlap condition requires row sparsity $ζ= Ω(\sqrt{\log r / \log \log r})$ to achieve linear-dimension injectivity, confirming that the $\sqrt{\log r}$ scale is fundamental. To bypass the lack of an operator-norm upper bound when $ζ= O(\sqrt{\log r})$, we decouple the sketch into two independent halves and introduce three techniques: (i) a uniform lower bound for flat vectors via randomized rounding, controlling discretization error using solely the Frobenius norm; (ii) a Gaussian truncation and minimax duality argument showing that all non-flat vectors escaping injectivity in the first half concentrate on a single universal coordinate subset $I \subseteq [n]$ of size $O(r/ζ)$, bypassing union bounds over supports; and (iii) a two-tier coordinate isolation analysis in the second half that suppresses collisions to $e^{-Ω(ζ^2)} \ll 1/r$, reliably preserving these heavy coordinates.

Authors: Lorenzo Beretta, Simone Bombari, Margalit Glasgow, Raphael Arkady Meyer, Kijun Shin, David P. Woodruff

An Oblivious Subspace Injection (OSI) is a random sketching matrix that reliably preserves the minimum singular value of any fixed subspace with high constant probability, requiring only a one-sided spectral lower bound without controlling upper singular values. While classical Oblivious Subspace Embeddings (OSEs) fundamentally require row sparsity $Ω(\log r)$ at linear sketch dimension $k = O(r)$, we show that one-sided injectivity permits strictly sparser sketches, establishing matching (up to $\log\log r$ factors) upper and lower bounds of $\tildeΘ(\sqrt{\log r})$. First, we prove that the SparseStack sketch achieves constant injectivity at linear dimension $k = Θ(r)$ with row sparsity only $ζ= O(\sqrt{\log r})$, well below the $Θ(\log r)$ barrier where standard operator-norm concentration fails. Second, we establish a matching lower bound: any isotropic random sketching matrix satisfying a mild row-overlap condition requires row sparsity $ζ= Ω(\sqrt{\log r / \log \log r})$ to achieve linear-dimension injectivity, confirming that the $\sqrt{\log r}$ scale is fundamental. To bypass the lack of an operator-norm upper bound when $ζ= O(\sqrt{\log r})$, we decouple the sketch into two independent halves and introduce three techniques: (i) a uniform lower bound for flat vectors via randomized rounding, controlling discretization error using solely the Frobenius norm; (ii) a Gaussian truncation and minimax duality argument showing that all non-flat vectors escaping injectivity in the first half concentrate on a single universal coordinate subset $I \subseteq [n]$ of size $O(r/ζ)$, bypassing union bounds over supports; and (iii) a two-tier coordinate isolation analysis in the second half that suppresses collisions to $e^{-Ω(ζ^2)} \ll 1/r$, reliably preserving these heavy coordinates.

Tuesday, November 10 2026, 01:00

Flows in High Degree Places: Sublinear Spanning Tree Congestion

Authors: D Ellis Hershkowitz, Richard Z Huang

The congestion of a spanning tree $T$ of graph $G$ is the maximum congestion of an edge of $T$ if each edge of $G$ is routed on $T$. Prior work showed that obtaining a minimum congestion spanning tree is NP-hard to approximate within $6/5$, yet the only known approximation algorithm is a trivial $n/2$-approximation from returning any spanning tree of the graph. We give a simple poly-time $\tilde{O}(n^{6/7})$-approximation for minimum congestion spanning tree on $n$-vertex graphs. We obtain this result via a $\tilde{O}(m^{2/3})$-approximation, where $m$ is the number of edges. Our algorithm is based on a new \textit{flow-or-degree decomposition} which deletes a small number of edges so that each remaining component either has low degree or admits a low congestion flow. We also show that, assuming the Small Set Expansion Hypothesis, it is NP-hard to $O(1)$-approximate this problem, even on graphs of constant maximum degree.

Authors: D Ellis Hershkowitz, Richard Z Huang

The congestion of a spanning tree $T$ of graph $G$ is the maximum congestion of an edge of $T$ if each edge of $G$ is routed on $T$. Prior work showed that obtaining a minimum congestion spanning tree is NP-hard to approximate within $6/5$, yet the only known approximation algorithm is a trivial $n/2$-approximation from returning any spanning tree of the graph. We give a simple poly-time $\tilde{O}(n^{6/7})$-approximation for minimum congestion spanning tree on $n$-vertex graphs. We obtain this result via a $\tilde{O}(m^{2/3})$-approximation, where $m$ is the number of edges. Our algorithm is based on a new \textit{flow-or-degree decomposition} which deletes a small number of edges so that each remaining component either has low degree or admits a low congestion flow. We also show that, assuming the Small Set Expansion Hypothesis, it is NP-hard to $O(1)$-approximate this problem, even on graphs of constant maximum degree.

Tuesday, November 10 2026, 01:00

Learning Rate Decay Can Exponentially Accelerate SGD for Global Optimization of Nonconvex Functions

Authors: Guneykan Ozgul, Dylan Herman, Junhyung Lyle Kim, Shree Hari Sureshbabu, Shouvanik Chakrabarti

Stochastic Gradient Descent (SGD) is a workhorse algorithm for continuous optimization. It is a folk belief among deep learning practitioners that learning rate decay in SGD substantially improves performance. However, it has remained unclear whether learning rate decay offers an asymptotic advantage for global nonconvex optimization. We answer this question in the affirmative by demonstrating a natural class of nonconvex functions for which, we prove that SGD with learning rate decay requires exponentially fewer gradient queries for global optimization than SGD with any fixed learning rate in the small-step size regime. To the best of our knowledge, this is the first exponential advantage that has been demonstrated for learning rate decay. The class of functions we consider includes many popular benchmark nonconvex functions. Our results provide a robust explanatory theory for the empirically observed benefits of learning rate decay for global optimization. We show that annealing performs the non-logconcave sampling task associated to the optimization problem in polynomial time, via a novel application of weak Poincaré inequalities.

Authors: Guneykan Ozgul, Dylan Herman, Junhyung Lyle Kim, Shree Hari Sureshbabu, Shouvanik Chakrabarti

Stochastic Gradient Descent (SGD) is a workhorse algorithm for continuous optimization. It is a folk belief among deep learning practitioners that learning rate decay in SGD substantially improves performance. However, it has remained unclear whether learning rate decay offers an asymptotic advantage for global nonconvex optimization. We answer this question in the affirmative by demonstrating a natural class of nonconvex functions for which, we prove that SGD with learning rate decay requires exponentially fewer gradient queries for global optimization than SGD with any fixed learning rate in the small-step size regime. To the best of our knowledge, this is the first exponential advantage that has been demonstrated for learning rate decay. The class of functions we consider includes many popular benchmark nonconvex functions. Our results provide a robust explanatory theory for the empirically observed benefits of learning rate decay for global optimization. We show that annealing performs the non-logconcave sampling task associated to the optimization problem in polynomial time, via a novel application of weak Poincaré inequalities.

Tuesday, November 10 2026, 01:00

Computing EF1 Allocations under Budget Constraints in Polynomial Time

Authors: Tingwei Hu, Lili Mei, Guochuan Zhang

We study the fair allocation of indivisible items to agents with budget constraints and additive valuations. Each agent compares its own bundle with the subsets of another bundle that fit its budget. An allocation is envy-free up to one item (EF1) if, for every agent and every nonempty subset under comparison, removing an item of maximum value to that agent eliminates its envy. We require EF1 both between agents and towards the charity, which contains all unallocated items. EF1 allocations exist, but computing them in polynomial time has remained open even for homogeneous valuations. We resolve this problem with a polynomial-time algorithm for EF1 allocations under heterogeneous valuations. We also prove that detecting an EF1 violation is NP-complete. Equivalently, checking whether a given allocation is EF1 is coNP-complete. Finally, we give polynomial-time algorithms for envy-freeness up to any item (EFX) both when each agent's valuation is proportional to item size and when each agent has a fixed number of distinct positive item values. As an application, for homogeneous valuations and any $\varepsilon>0$, we obtain an EF1 allocation with at least a $(1/2-\varepsilon)$ fraction of the maximum social welfare in time polynomial in the input length and $1/\varepsilon$.

Authors: Tingwei Hu, Lili Mei, Guochuan Zhang

We study the fair allocation of indivisible items to agents with budget constraints and additive valuations. Each agent compares its own bundle with the subsets of another bundle that fit its budget. An allocation is envy-free up to one item (EF1) if, for every agent and every nonempty subset under comparison, removing an item of maximum value to that agent eliminates its envy. We require EF1 both between agents and towards the charity, which contains all unallocated items. EF1 allocations exist, but computing them in polynomial time has remained open even for homogeneous valuations. We resolve this problem with a polynomial-time algorithm for EF1 allocations under heterogeneous valuations. We also prove that detecting an EF1 violation is NP-complete. Equivalently, checking whether a given allocation is EF1 is coNP-complete. Finally, we give polynomial-time algorithms for envy-freeness up to any item (EFX) both when each agent's valuation is proportional to item size and when each agent has a fixed number of distinct positive item values. As an application, for homogeneous valuations and any $\varepsilon>0$, we obtain an EF1 allocation with at least a $(1/2-\varepsilon)$ fraction of the maximum social welfare in time polynomial in the input length and $1/\varepsilon$.

Tuesday, November 10 2026, 01:00

No Exponential Quantum Advantage for Decoded Quantum Interferometry with Local Quantum Decoders

Authors: Brandon Augustino, Shouvanik Chakrabarti, Enrico Fontana, Dylan Herman, Guneykan Ozgul, Ruslan Shaydulin, Shreehari Sureshbabu, Jacob Watkins

Decoded quantum interferometry (DQI) reduces optimization of sparse linear constraint problems, and sampling from their Gibbs measures, to decoding a linear code from a quantum state. We study DQI on random $D$-regular $k$-sparse instances with $k\ge3$, for which local decoders such as belief propagation with quantum messages (BPQM) are commonly used. Such decoders estimate a typical codeword symbol using only the qudits within logarithmic distance of it. Our main insight is that the thresholds of the decoding problem induced by DQI correspond to phase transitions of the Gibbs measure that are well studied in statistical physics: tree reconstruction for local decoders and, for Max-$k$-XORSAT, condensation for the optimal decoder. This insight allows us to prove two main results. First, we prove an information-theoretic obstruction: no local decoder can recover the codeword with high probability above the tree reconstruction threshold. Conversely, we show that a decoder based on BPQM reaches this threshold. Since the cavity method predicts that local Markov chains equilibrate rapidly up to the same threshold, we expect DQI with local decoders to give no exponential speedup for sampling and no advantage in approximate optimization with amplitude functions that depend only on the number of violated constraints. Second, we consider the optimal decoder (the pretty-good measurement, for which no efficient implementation is known) and show that it succeeds exactly below the condensation threshold. The condensation threshold thus provides a tight upper bound on the performance of DQI with these amplitude functions, providing a strict refinement of the limit that Jordan et al.\ derived from the Holevo bound. Our results point to efficient nonlocal decoders that succeed between the two thresholds as the only viable path to a quantum advantage with DQI on these instances.

Authors: Brandon Augustino, Shouvanik Chakrabarti, Enrico Fontana, Dylan Herman, Guneykan Ozgul, Ruslan Shaydulin, Shreehari Sureshbabu, Jacob Watkins

Decoded quantum interferometry (DQI) reduces optimization of sparse linear constraint problems, and sampling from their Gibbs measures, to decoding a linear code from a quantum state. We study DQI on random $D$-regular $k$-sparse instances with $k\ge3$, for which local decoders such as belief propagation with quantum messages (BPQM) are commonly used. Such decoders estimate a typical codeword symbol using only the qudits within logarithmic distance of it. Our main insight is that the thresholds of the decoding problem induced by DQI correspond to phase transitions of the Gibbs measure that are well studied in statistical physics: tree reconstruction for local decoders and, for Max-$k$-XORSAT, condensation for the optimal decoder. This insight allows us to prove two main results. First, we prove an information-theoretic obstruction: no local decoder can recover the codeword with high probability above the tree reconstruction threshold. Conversely, we show that a decoder based on BPQM reaches this threshold. Since the cavity method predicts that local Markov chains equilibrate rapidly up to the same threshold, we expect DQI with local decoders to give no exponential speedup for sampling and no advantage in approximate optimization with amplitude functions that depend only on the number of violated constraints. Second, we consider the optimal decoder (the pretty-good measurement, for which no efficient implementation is known) and show that it succeeds exactly below the condensation threshold. The condensation threshold thus provides a tight upper bound on the performance of DQI with these amplitude functions, providing a strict refinement of the limit that Jordan et al.\ derived from the Holevo bound. Our results point to efficient nonlocal decoders that succeed between the two thresholds as the only viable path to a quantum advantage with DQI on these instances.

Tuesday, November 10 2026, 01:00

Better Bounds For Graph Reconstruction

Authors: Antoni Buraczewski, Paweł Gawrychowski

The goal in graph reconstruction problems is to learn an initially unknown graph using only a few queries of a specific type. Possibly the most natural query type is retrieving the distance between any two vertices of a connected, undirected, and unweighted graph. For a bounded-degree graph on $n$ vertices, Kannan, Mathieu, and Zhou [TALG 2018] showed a randomized algorithm that uses only $\tilde{\mathcal{O}}(n^{3/2})$ such queries. On the lower bound side, it is only known that any randomized algorithm needs $Ω(n\log n)$ queries. We make a significant step forward to bridging these bounds by designing a randomized algorithm that uses $n^{4/3+o(1)}$ queries. This requires extending the original approach based on Voronoi diagrams into a recursive procedure that partitions a cell into smaller cells. While such a recursive approach has been used in the related graph verification problem, in the more general graph reconstruction problem we need to operate without knowing the graph, which requires a new (and more technical) approach.

Authors: Antoni Buraczewski, Paweł Gawrychowski

The goal in graph reconstruction problems is to learn an initially unknown graph using only a few queries of a specific type. Possibly the most natural query type is retrieving the distance between any two vertices of a connected, undirected, and unweighted graph. For a bounded-degree graph on $n$ vertices, Kannan, Mathieu, and Zhou [TALG 2018] showed a randomized algorithm that uses only $\tilde{\mathcal{O}}(n^{3/2})$ such queries. On the lower bound side, it is only known that any randomized algorithm needs $Ω(n\log n)$ queries. We make a significant step forward to bridging these bounds by designing a randomized algorithm that uses $n^{4/3+o(1)}$ queries. This requires extending the original approach based on Voronoi diagrams into a recursive procedure that partitions a cell into smaller cells. While such a recursive approach has been used in the related graph verification problem, in the more general graph reconstruction problem we need to operate without knowing the graph, which requires a new (and more technical) approach.

Tuesday, November 10 2026, 01:00

Optimizing the Steady-State Travel Time of Nash Flows over Time with Capacity Increases

Authors: Niklas Dahlmeier, Paul Paschmanns, Laura Ritter, Ruben Skorupinski

We study budgeted capacity augmentation for single-source single-sink \NFOTs under constant inflow $u_0$. The goal is to increase capacities on arcs subject to a global budget constraint in order to minimize the steady-state travel time in the network. We show that this problem is weakly NP-hard on series-parallel networks and provide an exact pseudopolynomial algorithm as well as a fully polynomial algorithm that attains the optimum travel time for budget $B$ while spending at most $B + \varepsilon\cdot\max\{B,u_0\}$. On general networks the problem becomes strongly NP-hard. Moreover, unless $\text{P} = \text{NP}$, for every fixed $\varepsilon \in (0,1)$, no polynomial-time algorithm can achieve the optimal steady-state travel time while exceeding the budget by at most a factor of $n^{1-\varepsilon}$. Lastly, we give structural properties about steady-state travel time under varying arc capacities. We identify capacity thresholds determined by subsets of $s$-$t$-cuts and show that the travel time is constant in the interior of the cells resulting from the corresponding hyperplane arrangement.

Authors: Niklas Dahlmeier, Paul Paschmanns, Laura Ritter, Ruben Skorupinski

We study budgeted capacity augmentation for single-source single-sink \NFOTs under constant inflow $u_0$. The goal is to increase capacities on arcs subject to a global budget constraint in order to minimize the steady-state travel time in the network. We show that this problem is weakly NP-hard on series-parallel networks and provide an exact pseudopolynomial algorithm as well as a fully polynomial algorithm that attains the optimum travel time for budget $B$ while spending at most $B + \varepsilon\cdot\max\{B,u_0\}$. On general networks the problem becomes strongly NP-hard. Moreover, unless $\text{P} = \text{NP}$, for every fixed $\varepsilon \in (0,1)$, no polynomial-time algorithm can achieve the optimal steady-state travel time while exceeding the budget by at most a factor of $n^{1-\varepsilon}$. Lastly, we give structural properties about steady-state travel time under varying arc capacities. We identify capacity thresholds determined by subsets of $s$-$t$-cuts and show that the travel time is constant in the interior of the cells resulting from the corresponding hyperplane arrangement.

Tuesday, November 10 2026, 01:00

Sampling Colorings on $G(n, d/n)$ via Glauber Dynamics

Authors: Leon Schiller

We study the mixing time of Glauber dynamics for sampling uniformly from the proper $q$-colorings of sparse Erdős-Rényi random graphs $G(n,d/n)$. Our main result establishes that with probability $1-o(1)$ over the draw of $G$, Glauber dynamics is rapidly mixing for every $q\geq d+d^{1/2+\varepsilon}$, given any fixed $0<\varepsilon\leq1/2$ and any sufficiently large fixed $d$. In particular, the surplus of colors above $d$ is sublinear and close to $\sqrt{d}$. The achieved color range therefore approaches the tree uniqueness threshold and improves over the previously best achievable number of colors $q > 1.763 d$ due to Efthymiou, Hayes, Stefankovic, and Vigoda [SODA 2018]. Moreover, our mixing-time bound is $O(n^{2+o_d(1)})$, and hence almost quadratic as $d\rightarrow\infty$. Our proof is based on establishing a spectral gap for block dynamics resulting from a suitable decomposition of $G(n,d/n)$ into blocks and singleton vertices, ensuring that every high-degree vertex is encapsulated within a block. Our main technical contribution is a novel way of establishing rapid mixing of block dynamics via an integrated Bochner-Bakry-Émery approach, which we expect to have applications beyond sampling colorings. We further devise a new block decomposition of $G(n,d/n)$ that strengthens and simplifies similar constructions used in previous work. Finally, to compare block and Glauber dynamics, we establish a novel recursive separator decomposition of tree-like blocks, together with a coupling argument that avoids the need for nearly uniform color marginals at vertices adjacent to hubs. This comparison argument might be of independent interest.

Authors: Leon Schiller

We study the mixing time of Glauber dynamics for sampling uniformly from the proper $q$-colorings of sparse Erdős-Rényi random graphs $G(n,d/n)$. Our main result establishes that with probability $1-o(1)$ over the draw of $G$, Glauber dynamics is rapidly mixing for every $q\geq d+d^{1/2+\varepsilon}$, given any fixed $0<\varepsilon\leq1/2$ and any sufficiently large fixed $d$. In particular, the surplus of colors above $d$ is sublinear and close to $\sqrt{d}$. The achieved color range therefore approaches the tree uniqueness threshold and improves over the previously best achievable number of colors $q > 1.763 d$ due to Efthymiou, Hayes, Stefankovic, and Vigoda [SODA 2018]. Moreover, our mixing-time bound is $O(n^{2+o_d(1)})$, and hence almost quadratic as $d\rightarrow\infty$. Our proof is based on establishing a spectral gap for block dynamics resulting from a suitable decomposition of $G(n,d/n)$ into blocks and singleton vertices, ensuring that every high-degree vertex is encapsulated within a block. Our main technical contribution is a novel way of establishing rapid mixing of block dynamics via an integrated Bochner-Bakry-Émery approach, which we expect to have applications beyond sampling colorings. We further devise a new block decomposition of $G(n,d/n)$ that strengthens and simplifies similar constructions used in previous work. Finally, to compare block and Glauber dynamics, we establish a novel recursive separator decomposition of tree-like blocks, together with a coupling argument that avoids the need for nearly uniform color marginals at vertices adjacent to hubs. This comparison argument might be of independent interest.

Tuesday, November 10 2026, 01:00

How Hard Is Tokenisation? Algorithms and Complexity Beyond NP-Hardness

Authors: Andrei Robert Ion, Rares Mateiu, Adrian Miclaus, Alexandru Popa

Given a collection of strings and a vocabulary budget, how well can we minimize its total token count, and how does the answer change with the tokenisation model? For binary direct tokenisation, previous work proved APX-hardness but left constant-factor approximability unresolved \citep{kastreva2026}. We rule out every polynomial-time $M^{1-\varepsilon}$ approximation, for fixed $\varepsilon>0$, unless $\mathrm{P}=\mathrm{NP}$, even on distinct equal-length strings with $K=m-1$. Here $M$ is the total input length and $m$ is the number of input strings. We complement this lower bound with a deterministic polynomial-time $O(M/\log M)$ approximation algorithm and stronger bounds for selected vocabulary budgets. For the different objective of maximizing compression savings, GreedTok was shown to attain a constant fraction of the optimum on unrestricted alphabets \citep{baliga2026}; we establish a deterministic EPTAS for direct savings on every fixed alphabet and analyze greedy selection by true marginal gains. For exact optimization, we give a binary direct algorithm running in $2^{O(n\log\log n/\log n)}\poly(I)$ time, where $n$ is the total length of the distinct input strings and $I$ is the input encoding length. Finally, we resolve two unary merge-model hardness questions left by the bounded-alphabet study \citep{kastreva2026}: standard unary bottom-up tokenisation is APX-hard, and unary optimal pair encoding is strongly NP-complete, strengthening the previously known weak hardness.

Authors: Andrei Robert Ion, Rares Mateiu, Adrian Miclaus, Alexandru Popa

Given a collection of strings and a vocabulary budget, how well can we minimize its total token count, and how does the answer change with the tokenisation model? For binary direct tokenisation, previous work proved APX-hardness but left constant-factor approximability unresolved \citep{kastreva2026}. We rule out every polynomial-time $M^{1-\varepsilon}$ approximation, for fixed $\varepsilon>0$, unless $\mathrm{P}=\mathrm{NP}$, even on distinct equal-length strings with $K=m-1$. Here $M$ is the total input length and $m$ is the number of input strings. We complement this lower bound with a deterministic polynomial-time $O(M/\log M)$ approximation algorithm and stronger bounds for selected vocabulary budgets. For the different objective of maximizing compression savings, GreedTok was shown to attain a constant fraction of the optimum on unrestricted alphabets \citep{baliga2026}; we establish a deterministic EPTAS for direct savings on every fixed alphabet and analyze greedy selection by true marginal gains. For exact optimization, we give a binary direct algorithm running in $2^{O(n\log\log n/\log n)}\poly(I)$ time, where $n$ is the total length of the distinct input strings and $I$ is the input encoding length. Finally, we resolve two unary merge-model hardness questions left by the bounded-alphabet study \citep{kastreva2026}: standard unary bottom-up tokenisation is APX-hard, and unary optimal pair encoding is strongly NP-complete, strengthening the previously known weak hardness.

Tuesday, November 10 2026, 01:00

A 1.764-competitive algorithm for online fractional vertex cover on forests

Authors: Marcin Bienkowski

We study online fractional vertex cover on forests under edge arrival. We give a deterministic algorithm with competitive ratio at most $4 - \sqrt{5} \approx 1.764$, which simultaneously produces an online fractional matching with competitive ratio of at least $1/(4 - \sqrt{5}) \approx 0.566$. This improves on the previous best competitive ratios of $11/6 \approx 1.833$ and $5/9 \approx 0.555$, respectively.

Authors: Marcin Bienkowski

We study online fractional vertex cover on forests under edge arrival. We give a deterministic algorithm with competitive ratio at most $4 - \sqrt{5} \approx 1.764$, which simultaneously produces an online fractional matching with competitive ratio of at least $1/(4 - \sqrt{5}) \approx 0.566$. This improves on the previous best competitive ratios of $11/6 \approx 1.833$ and $5/9 \approx 0.555$, respectively.

Tuesday, November 10 2026, 01:00

Tight Competitive Bounds for Online Matching on Forests

Authors: Bartłomiej Bosek, Grzegorz Gutowski, Paweł Putra, Anna Zych-Pawlewicz

We study the online matching problem, its fractional relaxation, and the fractional online vertex cover problem on forests whose edges arrive one at a time in an arbitrary order. Matching decisions are irrevocable, and vertex-cover weights can only increase over time. For online matching, we present a randomized algorithm with expected competitive ratio $4/7$ against an oblivious adversary, improving the previous $5/9$ guarantee. We also present a deterministic algorithm that simultaneously maintains a $4/7$-competitive fractional matching and a $7/4$-competitive fractional vertex cover. This deterministic algorithm improves the previous $5/9$ guarantee for fractional matching and the $11/6$ guarantee for fractional vertex cover. We prove that $4/7$ is optimal for both randomized and fractional matching, even on growing trees of maximum degree four.

Authors: Bartłomiej Bosek, Grzegorz Gutowski, Paweł Putra, Anna Zych-Pawlewicz

We study the online matching problem, its fractional relaxation, and the fractional online vertex cover problem on forests whose edges arrive one at a time in an arbitrary order. Matching decisions are irrevocable, and vertex-cover weights can only increase over time. For online matching, we present a randomized algorithm with expected competitive ratio $4/7$ against an oblivious adversary, improving the previous $5/9$ guarantee. We also present a deterministic algorithm that simultaneously maintains a $4/7$-competitive fractional matching and a $7/4$-competitive fractional vertex cover. This deterministic algorithm improves the previous $5/9$ guarantee for fractional matching and the $11/6$ guarantee for fractional vertex cover. We prove that $4/7$ is optimal for both randomized and fractional matching, even on growing trees of maximum degree four.

Tuesday, November 10 2026, 01:00

Instance-Optimality of Single-Pair Personalized PageRank

Authors: Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, Shuyi Yan

We consider discounted random walks in a directed simple graph $G=(V,E)$. Such a walk terminates at each step with a constant probability $α\in (0,1)$ and otherwise continues to a uniform random out-neighbor. Given a pair $(s,t)\in V^2$, we denote by $π(s,t)$ the probability of such a discounted random walk starting at $s$ terminates at $t$. This probability is referred to as the Personalized PageRank (PPR) of $t$ with respect to $s$. The problem of estimating $π(s,t)$ within a constant multiplicative factor with constant success probability for a given vertex pair $(s,t)$ is called the single-pair PPR estimation problem. We investigate the instance-optimality of single-pair PPR estimation in directed graphs. We present a new algorithm that aims at instance-optimal running time up to polylogarithmic factors. In other words, no correct algorithm can run faster than our algorithm on any valid graph by more than a polylogarithmic factor. We show that our algorithm is instance-optimal in the standard adjacency-list query model if $n$ is not known (as is often the case for most of the large networks to which PPR is applied). When $n$ is known, our algorithm is still instance-optimal for sparse graphs. For denser graphs, when $n$ is known, we show that no algorithm can achieve general instance-optimality. For the multiplicative approximation of PPR, we also improve the worst-case complexity from $O(mn\log n)$ to the near-optimal $O(m\log n)$. This is not only for the single-pair discussed above, but also for the single-source and single-target variants of the PPR estimation problem.

Authors: Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, Shuyi Yan

We consider discounted random walks in a directed simple graph $G=(V,E)$. Such a walk terminates at each step with a constant probability $α\in (0,1)$ and otherwise continues to a uniform random out-neighbor. Given a pair $(s,t)\in V^2$, we denote by $π(s,t)$ the probability of such a discounted random walk starting at $s$ terminates at $t$. This probability is referred to as the Personalized PageRank (PPR) of $t$ with respect to $s$. The problem of estimating $π(s,t)$ within a constant multiplicative factor with constant success probability for a given vertex pair $(s,t)$ is called the single-pair PPR estimation problem. We investigate the instance-optimality of single-pair PPR estimation in directed graphs. We present a new algorithm that aims at instance-optimal running time up to polylogarithmic factors. In other words, no correct algorithm can run faster than our algorithm on any valid graph by more than a polylogarithmic factor. We show that our algorithm is instance-optimal in the standard adjacency-list query model if $n$ is not known (as is often the case for most of the large networks to which PPR is applied). When $n$ is known, our algorithm is still instance-optimal for sparse graphs. For denser graphs, when $n$ is known, we show that no algorithm can achieve general instance-optimality. For the multiplicative approximation of PPR, we also improve the worst-case complexity from $O(mn\log n)$ to the near-optimal $O(m\log n)$. This is not only for the single-pair discussed above, but also for the single-source and single-target variants of the PPR estimation problem.

Tuesday, November 10 2026, 01:00

Longest Unbordered Factors in Linear Time

Authors: Takuya Mieno

A border of a string is a nonempty proper prefix that is also a suffix. A string is unbordered if it has no border. The length of a longest unbordered factor is closely related to the smallest period of the string. While the smallest period can be computed in linear time, the best previously known algorithm for computing a longest unbordered factor takes $O(n \log n)$ worst-case time for a string of length $n$. Whether this computational problem can be solved in worst-case linear time has remained open. In this paper, we resolve this open problem for integer alphabets of size $n^{O(1)}$ by presenting an algorithm that computes all occurrences of longest unbordered factors in $O(n)$ worst-case time. Our algorithm uses a combinatorial property of longest unbordered factors and scans the input string in both directions to capture all occurrences achieving the global maximum without computing the local maximum at every starting position.

Authors: Takuya Mieno

A border of a string is a nonempty proper prefix that is also a suffix. A string is unbordered if it has no border. The length of a longest unbordered factor is closely related to the smallest period of the string. While the smallest period can be computed in linear time, the best previously known algorithm for computing a longest unbordered factor takes $O(n \log n)$ worst-case time for a string of length $n$. Whether this computational problem can be solved in worst-case linear time has remained open. In this paper, we resolve this open problem for integer alphabets of size $n^{O(1)}$ by presenting an algorithm that computes all occurrences of longest unbordered factors in $O(n)$ worst-case time. Our algorithm uses a combinatorial property of longest unbordered factors and scans the input string in both directions to capture all occurrences achieving the global maximum without computing the local maximum at every starting position.

Tuesday, November 10 2026, 01:00

The Power of Migration on Identical Machines is Two

Authors: Nicole Megow, Bart Zondervan

We study the power of migration in preemptive scheduling on identical parallel machines with time windows. If an instance admits a migratory schedule on $m$ machines, how many machines are needed to schedule it without migration? Kalyanasundaram and Pruhs (2001) showed that $6m-5$ machines suffice and gave a polynomial-time construction using $12m-5$ machines, whereas $2m-1$ machines may be necessary. We close this gap, which has been open for more than 25 years: $2m-1$ machines always suffice, and a corresponding non-migratory schedule can be computed in strongly polynomial time. Our proof has two main technical ingredients. First, we introduce supercanonical schedules and prove that the conflict graph of every such schedule on $m$ machines is $(2m-1)$-colorable. The key property is that whenever a job pauses, the jobs running alongside it just before and just after the pause separate the jobs processed entirely during the pause from the rest of the conflict graph. We use these separators and Kempe exchanges to compute a coloring in polynomial time. Second, we construct a supercanonical schedule with polynomially many processing blocks by shrinking job windows as far as possible while preserving feasibility and solving a min-cost flow problem.

Authors: Nicole Megow, Bart Zondervan

We study the power of migration in preemptive scheduling on identical parallel machines with time windows. If an instance admits a migratory schedule on $m$ machines, how many machines are needed to schedule it without migration? Kalyanasundaram and Pruhs (2001) showed that $6m-5$ machines suffice and gave a polynomial-time construction using $12m-5$ machines, whereas $2m-1$ machines may be necessary. We close this gap, which has been open for more than 25 years: $2m-1$ machines always suffice, and a corresponding non-migratory schedule can be computed in strongly polynomial time. Our proof has two main technical ingredients. First, we introduce supercanonical schedules and prove that the conflict graph of every such schedule on $m$ machines is $(2m-1)$-colorable. The key property is that whenever a job pauses, the jobs running alongside it just before and just after the pause separate the jobs processed entirely during the pause from the rest of the conflict graph. We use these separators and Kempe exchanges to compute a coloring in polynomial time. Second, we construct a supercanonical schedule with polynomially many processing blocks by shrinking job windows as far as possible while preserving feasibility and solving a min-cost flow problem.

Tuesday, November 10 2026, 01:00

Tight Bounds for Sublinear Metric Steiner Tree

Authors: Sepideh Mahabadi, Mohammad Roghani, Jakub Tarnawski, Ali Vakilian

In the metric Steiner tree problem we are given an $n$-point metric space $(V,w)$ and a set $T\subseteq V$ of terminals, and we seek the smallest weight $\mathrm{OPT}$ of a tree that spans $T$ and may use points of $V\setminus T$. We study sublinear algorithms in the distance-oracle model, where the algorithm knows $V$ and $T$ upfront and can query the distance function $w$. The objective is to estimate $\mathrm{OPT}$ while querying significantly fewer than $n^2$ distances. The minimum spanning tree on $T$ estimates $\mathrm{OPT}$ within a factor $2$, and its weight can be approximated with $\widetilde{O}(n)$ queries. The natural question that has been studied is therefore the query complexity of estimating $\mathrm{OPT}$ within a factor strictly below $2$. Chen, Khanna, and Tan [SODA 2023] gave such an algorithm using $\widetilde{O}(n^{13/7})$ queries, and a lower bound of $\widetildeΩ(n^{6/5})$. Later Mahabadi, Roghani, Tarnawski, and Vakilian [ITCS 2025] improved the algorithm to $\widetilde{O}(n^{5/3})$, and Mahabadi, Tarnawski, and Vakilian [ITCS 2027] improved the lower bound to $\widetildeΩ(n^{7/5})$. In this work we settle the above question. We give an algorithm that estimates $\mathrm{OPT}$ within a factor $2-η_0$, for an absolute constant $η_0>0$, with query (and time) complexity $\widetilde{O}(n^{3/2})$. We also show that the exponent $3/2$ is tight.

Authors: Sepideh Mahabadi, Mohammad Roghani, Jakub Tarnawski, Ali Vakilian

In the metric Steiner tree problem we are given an $n$-point metric space $(V,w)$ and a set $T\subseteq V$ of terminals, and we seek the smallest weight $\mathrm{OPT}$ of a tree that spans $T$ and may use points of $V\setminus T$. We study sublinear algorithms in the distance-oracle model, where the algorithm knows $V$ and $T$ upfront and can query the distance function $w$. The objective is to estimate $\mathrm{OPT}$ while querying significantly fewer than $n^2$ distances. The minimum spanning tree on $T$ estimates $\mathrm{OPT}$ within a factor $2$, and its weight can be approximated with $\widetilde{O}(n)$ queries. The natural question that has been studied is therefore the query complexity of estimating $\mathrm{OPT}$ within a factor strictly below $2$. Chen, Khanna, and Tan [SODA 2023] gave such an algorithm using $\widetilde{O}(n^{13/7})$ queries, and a lower bound of $\widetildeΩ(n^{6/5})$. Later Mahabadi, Roghani, Tarnawski, and Vakilian [ITCS 2025] improved the algorithm to $\widetilde{O}(n^{5/3})$, and Mahabadi, Tarnawski, and Vakilian [ITCS 2027] improved the lower bound to $\widetildeΩ(n^{7/5})$. In this work we settle the above question. We give an algorithm that estimates $\mathrm{OPT}$ within a factor $2-η_0$, for an absolute constant $η_0>0$, with query (and time) complexity $\widetilde{O}(n^{3/2})$. We also show that the exponent $3/2$ is tight.

Tuesday, November 10 2026, 01:00

How Far Must a Test Explore? Certifying Composable Read-Write Interfaces from Their Own Random Edits

Authors: Jiaxuan Guo, Xingyu Ren, Jingxin Yang, Youran Sun, Haizhao Yang

Random tests of stateful code check laws, like reading back a written value, at states that random writes reach. The laws hold at every state exactly when the code behaves like a record with hidden data, so a pass is taken as evidence. This can fail: the laws can hold at every sample while the code is far from any record. The laws are local, but behaving like a record is global. Self-testing and stability theorems bridge this gap when the query distribution is controlled by the tester or preserved by the tested maps, but here it is generated by the tested writes, which need not be invertible. We ask how far a test must explore. An interface has $m$ binary variables, each with a read and two writes; in a record, reads return their bits and writes set only their bits. Samples are the ends of geometric runs of mean $\ell$ of uniformly random writes from a start distribution; a tester adds its own write chains of depth $d$. The distance is the least, over records with hidden data, of the largest probability that a read or write differs from it at a sample. From one start state, law checks at $O(m^2\log m/ε)$ samples certify, at fixed confidence, any threshold $ε$ after geometric runs of every positive mean, while runs of fixed positive length can pass at distance $1/2$. For the worst start distribution, take $ε=m^{-β}$ with fixed $0<β\le1$. For $d\le cm$ with fixed $c<1/2$, a polynomial budget of samples and writes exists exactly when $\ell=Ω(m)$, where law checks suffice. With no runs, the least depth for this budget is $(1/2+o(1))m$. Counting samples alone, in the worst case at $ε=1/(32m)$ and $d\le m/16$, the critical mean for a polynomial budget is $Θ(m\log(m/d)/\log m)$, by exhaustive search. So one start state needs only law checks, and the worst case at these thresholds needs exploration linear in $m$ if writes are counted.

Authors: Jiaxuan Guo, Xingyu Ren, Jingxin Yang, Youran Sun, Haizhao Yang

Random tests of stateful code check laws, like reading back a written value, at states that random writes reach. The laws hold at every state exactly when the code behaves like a record with hidden data, so a pass is taken as evidence. This can fail: the laws can hold at every sample while the code is far from any record. The laws are local, but behaving like a record is global. Self-testing and stability theorems bridge this gap when the query distribution is controlled by the tester or preserved by the tested maps, but here it is generated by the tested writes, which need not be invertible. We ask how far a test must explore. An interface has $m$ binary variables, each with a read and two writes; in a record, reads return their bits and writes set only their bits. Samples are the ends of geometric runs of mean $\ell$ of uniformly random writes from a start distribution; a tester adds its own write chains of depth $d$. The distance is the least, over records with hidden data, of the largest probability that a read or write differs from it at a sample. From one start state, law checks at $O(m^2\log m/ε)$ samples certify, at fixed confidence, any threshold $ε$ after geometric runs of every positive mean, while runs of fixed positive length can pass at distance $1/2$. For the worst start distribution, take $ε=m^{-β}$ with fixed $0<β\le1$. For $d\le cm$ with fixed $c<1/2$, a polynomial budget of samples and writes exists exactly when $\ell=Ω(m)$, where law checks suffice. With no runs, the least depth for this budget is $(1/2+o(1))m$. Counting samples alone, in the worst case at $ε=1/(32m)$ and $d\le m/16$, the critical mean for a polynomial budget is $Θ(m\log(m/d)/\log m)$, by exhaustive search. So one start state needs only law checks, and the worst case at these thresholds needs exploration linear in $m$ if writes are counted.

Tuesday, November 10 2026, 01:00

Internal Queries for Maximal Closed Substrings

Authors: Simone Faro, Francesco Pio Marino, Arianna Pavone

Searching for repetitions inside a selected part of a text raises a subtle boundary problem: removing neighbouring characters can create new maximal answers. We study this effect for maximal closed substrings (MCSs). A string is closed if it is a single letter or has a nonempty proper prefix occurring exactly twice, as its prefix and suffix; a closed occurrence is maximal if neither available one-character extension is closed. Previous work established $O(n\log n)$ bounds on the global MCS families of a length-$n$ text and developed enumeration algorithms. Internal queries instead ask for the MCSs of a chosen substring, with maximality evaluated inside that substring. Separating boundary occurrences from interior ones gives an $O(n\log n)$-word index that reports the $c$ answers in $O(\log n+c)$ time and supports counting and longest queries in $O(\log n)$ time. Our main result reduces reporting space to $O(n)$ words, with $O((c+1)\log^{1+\varepsilon} n)$ query time and $O(n\log^{1+\varepsilon} n)$ construction time for any fixed $\varepsilon>0$. Thus, after preprocessing, reporting requires no additive scan of the queried substring, which is particularly useful for long queries with few answers. Independently, we prove that at most $3(s-1)$ MCS occurrences of any length-$s$ substring cross a fixed gap between consecutive characters, with maximality evaluated inside that substring. The same bound holds for global MCSs contained in a length-$s$ block. A family over two letters attains $\lfloor(s-1)/2\rfloor$ crossings for every $s\ge3$, showing that the linear order of this local density bound is optimal.

Authors: Simone Faro, Francesco Pio Marino, Arianna Pavone

Searching for repetitions inside a selected part of a text raises a subtle boundary problem: removing neighbouring characters can create new maximal answers. We study this effect for maximal closed substrings (MCSs). A string is closed if it is a single letter or has a nonempty proper prefix occurring exactly twice, as its prefix and suffix; a closed occurrence is maximal if neither available one-character extension is closed. Previous work established $O(n\log n)$ bounds on the global MCS families of a length-$n$ text and developed enumeration algorithms. Internal queries instead ask for the MCSs of a chosen substring, with maximality evaluated inside that substring. Separating boundary occurrences from interior ones gives an $O(n\log n)$-word index that reports the $c$ answers in $O(\log n+c)$ time and supports counting and longest queries in $O(\log n)$ time. Our main result reduces reporting space to $O(n)$ words, with $O((c+1)\log^{1+\varepsilon} n)$ query time and $O(n\log^{1+\varepsilon} n)$ construction time for any fixed $\varepsilon>0$. Thus, after preprocessing, reporting requires no additive scan of the queried substring, which is particularly useful for long queries with few answers. Independently, we prove that at most $3(s-1)$ MCS occurrences of any length-$s$ substring cross a fixed gap between consecutive characters, with maximality evaluated inside that substring. The same bound holds for global MCSs contained in a length-$s$ block. A family over two letters attains $\lfloor(s-1)/2\rfloor$ crossings for every $s\ge3$, showing that the linear order of this local density bound is optimal.

Tuesday, November 10 2026, 01:00

Induced-Minor-Free Graphs Admit a Star-Based Separator of Size $\tilde{O}(\sqrt{n})$

Authors: Arnold Filtser, Nikolai Karol, Hung Le, Florian Lehner

Given an $H$-induced-minor-free graph $G$ with $n$ vertices, we construct a balanced separator of $G$ consisting of $\tilde{O}(|V(H)|\sqrt{n})$ star subgraphs. Our result is an induced-minor-free analog of the famous separator theorem for $H$-minor-free graphs by Alon, Seymour, and Thomas [J. Amer. Math. Soc., 1990]. A key idea in our construction is a new decomposition of $H$-induced-minor-free graphs, analogous to the cop decomposition for $H$-minor-free graphs by Andreae [J. Combin. Theory Ser. B, 1986]. We also discuss several algorithmic applications of our result.

Authors: Arnold Filtser, Nikolai Karol, Hung Le, Florian Lehner

Given an $H$-induced-minor-free graph $G$ with $n$ vertices, we construct a balanced separator of $G$ consisting of $\tilde{O}(|V(H)|\sqrt{n})$ star subgraphs. Our result is an induced-minor-free analog of the famous separator theorem for $H$-minor-free graphs by Alon, Seymour, and Thomas [J. Amer. Math. Soc., 1990]. A key idea in our construction is a new decomposition of $H$-induced-minor-free graphs, analogous to the cop decomposition for $H$-minor-free graphs by Andreae [J. Combin. Theory Ser. B, 1986]. We also discuss several algorithmic applications of our result.

Tuesday, November 10 2026, 01:00

Õptimal Facial Distance Labeling in Planar Graph

Authors: Noy Rotbart, Oren Weimann

We show how to assign labels of size $\tilde O(n^{1/3})$ to the vertices of a face $F$ in an $n$-vertex unweighted undirected planar graph $G$ s.t one can deduce the distance between any two vertices of $F$ given their labels alone. This matches (up to log factors) the $Ω(n^{1/3})$ lower bound of Gavoille, Peleg, Pérennes, Raz from 2004. Prior to our work, the smallest labels were of size $O(n^{1/2})$.

Authors: Noy Rotbart, Oren Weimann

We show how to assign labels of size $\tilde O(n^{1/3})$ to the vertices of a face $F$ in an $n$-vertex unweighted undirected planar graph $G$ s.t one can deduce the distance between any two vertices of $F$ given their labels alone. This matches (up to log factors) the $Ω(n^{1/3})$ lower bound of Gavoille, Peleg, Pérennes, Raz from 2004. Prior to our work, the smallest labels were of size $O(n^{1/2})$.

Tuesday, November 10 2026, 01:00

Tight B-Trees for Adjacency Labeling

Authors: Noy Rotbart

We show that planar graphs admit adjacency labels of size $\log n+O(\log\log n)$: every vertex gets a label of this many bits, and the adjacency of two vertices is decided from their labels alone, in constant time. This improves the $\log n+O(\sqrt{\log n\log\log n})$ of Dujmović et al. (J. ACM 2021) and the $\log n+O(\sqrt{\log n})$ of Gawrychowski and Janczewski (SOSA 2022). Equivalently, $n$-vertex planar graphs have an induced-universal graph with $n(\log n)^{O(1)}$ vertices, down from $n2^{O(\sqrt{\log n})}$. The same holds for graphs of bounded genus, apex-minor-free graphs and $k$-planar graphs. For every proper minor-closed class we get $\log n+O(\sqrt{\log n\log\log n})$, down from the $\log n+O((\log n)^{3/4})$ of Dujmović et al. (2026). Both results come from tight B-trees: nodes are split into parts of nearly equal size, so every node but the topmost stays within a factor $1\pm O(1/\log n)$ of its target size, and a root-to-leaf path loses only $O(\log\log n)$ bits.

Authors: Noy Rotbart

We show that planar graphs admit adjacency labels of size $\log n+O(\log\log n)$: every vertex gets a label of this many bits, and the adjacency of two vertices is decided from their labels alone, in constant time. This improves the $\log n+O(\sqrt{\log n\log\log n})$ of Dujmović et al. (J. ACM 2021) and the $\log n+O(\sqrt{\log n})$ of Gawrychowski and Janczewski (SOSA 2022). Equivalently, $n$-vertex planar graphs have an induced-universal graph with $n(\log n)^{O(1)}$ vertices, down from $n2^{O(\sqrt{\log n})}$. The same holds for graphs of bounded genus, apex-minor-free graphs and $k$-planar graphs. For every proper minor-closed class we get $\log n+O(\sqrt{\log n\log\log n})$, down from the $\log n+O((\log n)^{3/4})$ of Dujmović et al. (2026). Both results come from tight B-trees: nodes are split into parts of nearly equal size, so every node but the topmost stays within a factor $1\pm O(1/\log n)$ of its target size, and a root-to-leaf path loses only $O(\log\log n)$ bits.

Tuesday, November 10 2026, 01:00

Linear Set Reconciliation with $O(d)$ Additive Redundancy

Authors: Tianren Liu, Rui Xu, Tong Yang, Kangyang Zhou

In set reconciliation, two parties holding sets $A, B \subseteq [n]$ with $|A \triangle B| \le d$ seek to recover their symmetric difference from short $\mathbb{F}_2$-linear sketches, which is equivalent to recovering a $d$-sparse vector $x = \mathbf{1}_{A \triangle B} \in \mathbb{F}_2^n$ from its syndrome $Hx \in \mathbb{F}_2^m$. While the information-theoretic counting bound requires $L(n,d) := \log \sum_{i=0}^d \binom{n}{i} = d\log(n/d) + Θ(d)$ bits, existing polynomial-time schemes either use at least $d\lceil\log(n+1)\rceil$ bits (algebraic sketches, leaving an $Ω(d\log d)$-bit gap) or incur a multiplicative $(1+δ)L(n,d)$ overhead (peeling-based sketches), raising the question of whether $L(n,d) + O(d)$ bits is achievable with fast updates and decoding. We construct a public-coin randomized $\mathbb{F}_2$-linear sketch of length $L(n,d) + O(d) + λ\log^{O(1)} n$ that recovers each fixed input of weight at most $d$ with probability at least $1 - 2^{-λ}$, with $\log^{O(1)} n$ update time and $d\exp(O((\log\log n)^2)) = dn^{o(1)}$ worst-case decoding time. In particular, for $n \le 2^{\operatorname{polylog}(d)}$ and $λ\le d^{1-Ω(1)}$, this achieves $L(n,d) + O(d)$ bits with $\operatorname{polylog}(d)$ update time and $d^{1+o(1)}$ decoding time. Our construction encodes $d$ permuted rows via randomly scaled Pinsketch syndromes and sketches the resulting bit array column by column. Decoding columns sequentially from prefix predictions cancels recovered rows and exposes undecided rows as decoder-side hints, reducing each column to a hinted sparse recovery problem that we solve via a recursive sketch, HintSketch.

Authors: Tianren Liu, Rui Xu, Tong Yang, Kangyang Zhou

In set reconciliation, two parties holding sets $A, B \subseteq [n]$ with $|A \triangle B| \le d$ seek to recover their symmetric difference from short $\mathbb{F}_2$-linear sketches, which is equivalent to recovering a $d$-sparse vector $x = \mathbf{1}_{A \triangle B} \in \mathbb{F}_2^n$ from its syndrome $Hx \in \mathbb{F}_2^m$. While the information-theoretic counting bound requires $L(n,d) := \log \sum_{i=0}^d \binom{n}{i} = d\log(n/d) + Θ(d)$ bits, existing polynomial-time schemes either use at least $d\lceil\log(n+1)\rceil$ bits (algebraic sketches, leaving an $Ω(d\log d)$-bit gap) or incur a multiplicative $(1+δ)L(n,d)$ overhead (peeling-based sketches), raising the question of whether $L(n,d) + O(d)$ bits is achievable with fast updates and decoding. We construct a public-coin randomized $\mathbb{F}_2$-linear sketch of length $L(n,d) + O(d) + λ\log^{O(1)} n$ that recovers each fixed input of weight at most $d$ with probability at least $1 - 2^{-λ}$, with $\log^{O(1)} n$ update time and $d\exp(O((\log\log n)^2)) = dn^{o(1)}$ worst-case decoding time. In particular, for $n \le 2^{\operatorname{polylog}(d)}$ and $λ\le d^{1-Ω(1)}$, this achieves $L(n,d) + O(d)$ bits with $\operatorname{polylog}(d)$ update time and $d^{1+o(1)}$ decoding time. Our construction encodes $d$ permuted rows via randomly scaled Pinsketch syndromes and sketches the resulting bit array column by column. Decoding columns sequentially from prefix predictions cancels recovered rows and exposes undecided rows as decoder-side hints, reducing each column to a hinted sparse recovery problem that we solve via a recursive sketch, HintSketch.

Tuesday, November 10 2026, 01:00

Algorithms for Soft Tree Containment on Tree-Child Networks

Authors: Erdem Ugurlu, Louxin Zhang

Phylogenetic networks generalize evolutionary trees by allowing lineages to merge. The Tree Containment problem asks whether a network $N$ displays a tree $T$ on the same taxa. When the polytomies of $T$ reflect uncertainty rather than simultaneous divergence, the relevant notion is soft display, in which $N$ may refine the polytomies of $T$. We take it in its asymmetric reading. Soft Tree Containment is NP-hard on reticulation-visible networks even at reticulation indegree three, and it was open for a non-binary tree and a tree-child network with multifurcating tree nodes or reticulations of indegree above two. We solve this case. A cluster characterization reduces soft display to realizing all clusters of $T$ by one switching, and in a tree-child network the vertices that can realize a cluster lie on a single chain. The per-cluster requirements then form a constraint network of width one, which arc consistency decides without search, and a $2$-SAT instance when $N$ is binary. This yields, to our knowledge, the first polynomial-time algorithm for soft Tree Containment on single-labelled tree-child networks of arbitrary reticulation indegree, and a linear-time algorithm when $T$ is binary. The same method decides the symmetric variant of the problem, in which the polytomies of $N$ may be resolved as well.

Authors: Erdem Ugurlu, Louxin Zhang

Phylogenetic networks generalize evolutionary trees by allowing lineages to merge. The Tree Containment problem asks whether a network $N$ displays a tree $T$ on the same taxa. When the polytomies of $T$ reflect uncertainty rather than simultaneous divergence, the relevant notion is soft display, in which $N$ may refine the polytomies of $T$. We take it in its asymmetric reading. Soft Tree Containment is NP-hard on reticulation-visible networks even at reticulation indegree three, and it was open for a non-binary tree and a tree-child network with multifurcating tree nodes or reticulations of indegree above two. We solve this case. A cluster characterization reduces soft display to realizing all clusters of $T$ by one switching, and in a tree-child network the vertices that can realize a cluster lie on a single chain. The per-cluster requirements then form a constraint network of width one, which arc consistency decides without search, and a $2$-SAT instance when $N$ is binary. This yields, to our knowledge, the first polynomial-time algorithm for soft Tree Containment on single-labelled tree-child networks of arbitrary reticulation indegree, and a linear-time algorithm when $T$ is binary. The same method decides the symmetric variant of the problem, in which the polytomies of $N$ may be resolved as well.

Tuesday, November 10 2026, 01:00

Fast Approximate $\ell_p$ Chamfer Distance via Lopsided Embeddings and Structured JL

Authors: Ying Feng, David P. Woodruff

For two $d$-dimensional point sets $A, B$ of size $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} \mathsf{dist}(a, b)$, where $\mathsf{dist}$ is some underlying distance measure. We present efficient algorithms for approximating the Chamfer distance under the $\ell_p$ norm for $1 < p \le 2$. For the general case $1 < p < 2$, we utilize lopsided $\ell_p$ to $\ell_1$ embeddings with weak guarantees. We show that they are sufficient for preserving the Chamfer distance. In the high-dimensional regime, we use Fast Matrix Multiplication techniques to speed up the lopsided embeddings. These lead to $O(nd\cdot \mathrm{poly}(\log \log n, \log(1/\varepsilon))/\varepsilon^2)$ total time. For the specific Euclidean case ($p=2$), we leverage the Fast Johnson-Lindenstrauss Transform based on Toeplitz matrices and re-analyze the previous $\ell_1$-specific Chamfer algorithm in $\ell_2$. These achieve a runtime matching the recent state-of-the-art $\ell_1$ result of $O(nd(\log \log n + \log (1/\varepsilon))/\varepsilon^2)$, improving upon the previous $O(nd \log n/\varepsilon^2)$ bound for $\ell_2$. We also give additional results for the angular distance and upper and lower bounds in the streaming setting.

Authors: Ying Feng, David P. Woodruff

For two $d$-dimensional point sets $A, B$ of size $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} \mathsf{dist}(a, b)$, where $\mathsf{dist}$ is some underlying distance measure. We present efficient algorithms for approximating the Chamfer distance under the $\ell_p$ norm for $1 < p \le 2$. For the general case $1 < p < 2$, we utilize lopsided $\ell_p$ to $\ell_1$ embeddings with weak guarantees. We show that they are sufficient for preserving the Chamfer distance. In the high-dimensional regime, we use Fast Matrix Multiplication techniques to speed up the lopsided embeddings. These lead to $O(nd\cdot \mathrm{poly}(\log \log n, \log(1/\varepsilon))/\varepsilon^2)$ total time. For the specific Euclidean case ($p=2$), we leverage the Fast Johnson-Lindenstrauss Transform based on Toeplitz matrices and re-analyze the previous $\ell_1$-specific Chamfer algorithm in $\ell_2$. These achieve a runtime matching the recent state-of-the-art $\ell_1$ result of $O(nd(\log \log n + \log (1/\varepsilon))/\varepsilon^2)$, improving upon the previous $O(nd \log n/\varepsilon^2)$ bound for $\ell_2$. We also give additional results for the angular distance and upper and lower bounds in the streaming setting.

Tuesday, November 10 2026, 01:00

Online Correlated Exponential Clocks

Authors: Zhiyi Huang, Kalen Patton, Peilin Yang, Jiahao Zhao

In online resource allocation, we seek to allocate an incoming sequence of requests $j \in J$ among a set of offline resources $i \in I$, where decisions must be made immediately without knowledge of the future. State-of-the-art results typically follow from relax-and-round methods. For many problems, recent advances have come from improved methods of online rounding with negative correlations. However, such rounding schemes have largely been implemented on a case-by-case basis, making it difficult to transfer improvements and new ideas between problem settings. This leads us to ask the question: Can we develop unified methods of online rounding with negative correlations which can obtain state-of-the-art guarantees across different resource allocation settings? We answer this question affirmatively by introducing the framework of Online Correlated Exponential Clocks (OCEC). In this framework, fractional allocations $x_{ij}$ are assigned exponential clocks $Y_{ij} \sim \mathrm{Exp}(x_{ij})$, and each request $j$ is allocated to the resource $i$ with smallest $Y_{ij}$. The key idea is that the sequence $(Y_{ij})_j$ for each fixed $i$ is negatively correlated via an OCEC generator, resulting in better-than-baseline guarantees. Our contributions are two-fold: First, we introduce OCEC rounding as a problem-independent interface which any online algorithm can use. This isolates the challenge of online rounding into the self-contained problem of designing an effective OCEC generator. Second, we demonstrate the flexibility of this method by giving a single OCEC generator which yields improved competitive bounds in three distinct problems: 0.556 for Display Ads, 0.530 for AdWords, and 4.69 for squared-load balancing. Previous best results were obtained by Blanc and Charikar (FOCS 2021), Chen, Huang, and Sun (FOCS 2024), and Borst and Kashaev (SODA 2027), respectively.

Authors: Zhiyi Huang, Kalen Patton, Peilin Yang, Jiahao Zhao

In online resource allocation, we seek to allocate an incoming sequence of requests $j \in J$ among a set of offline resources $i \in I$, where decisions must be made immediately without knowledge of the future. State-of-the-art results typically follow from relax-and-round methods. For many problems, recent advances have come from improved methods of online rounding with negative correlations. However, such rounding schemes have largely been implemented on a case-by-case basis, making it difficult to transfer improvements and new ideas between problem settings. This leads us to ask the question: Can we develop unified methods of online rounding with negative correlations which can obtain state-of-the-art guarantees across different resource allocation settings? We answer this question affirmatively by introducing the framework of Online Correlated Exponential Clocks (OCEC). In this framework, fractional allocations $x_{ij}$ are assigned exponential clocks $Y_{ij} \sim \mathrm{Exp}(x_{ij})$, and each request $j$ is allocated to the resource $i$ with smallest $Y_{ij}$. The key idea is that the sequence $(Y_{ij})_j$ for each fixed $i$ is negatively correlated via an OCEC generator, resulting in better-than-baseline guarantees. Our contributions are two-fold: First, we introduce OCEC rounding as a problem-independent interface which any online algorithm can use. This isolates the challenge of online rounding into the self-contained problem of designing an effective OCEC generator. Second, we demonstrate the flexibility of this method by giving a single OCEC generator which yields improved competitive bounds in three distinct problems: 0.556 for Display Ads, 0.530 for AdWords, and 4.69 for squared-load balancing. Previous best results were obtained by Blanc and Charikar (FOCS 2021), Chen, Huang, and Sun (FOCS 2024), and Borst and Kashaev (SODA 2027), respectively.

Tuesday, November 10 2026, 01:00

Optimal Polynomial-Query Approximation for Consistent Submodular Maximization with Short Lookahead

Authors: Vaneet Aggarwal

Elements arrive one at a time, and after each arrival an algorithm holds a set of at most $k$ arrived elements. Its goal is a held set whose value under a monotone submodular function is close to the best value that $k$ arrived elements can achieve. The recourse of an update is the number of elements inserted into or removed from the held set, and we require it to be bounded independently of $k$ and of the number $n$ of arrivals. With evaluations only on sets of arrived elements and polynomially many of them, the best ratio a randomized algorithm can achieve is $2-\sqrt{2}\approx0.5858$. We show that seeing future arrivals raises the best ratio for polynomially many evaluations to $1-1/e\approx0.6321$, for deterministic and for randomized algorithms; no algorithm with polynomially many evaluations exceeds $1-1/e$, even when all elements are available at once. For every $\varepsilon>0$, a deterministic algorithm that sees the next $\lceil\varepsilon k/3\rceil$ arrivals keeps a set of value at least $1-1/e-\varepsilon$ times the optimum at every time, changes $O(\varepsilon^{-3})$ elements per arrival and makes $(n+1)^{O(\varepsilon^{-2})}$ evaluations. With the whole arrival sequence known in advance, a deterministic algorithm attains the same ratio with $O(\varepsilon^{-2})$ changes per arrival. A randomized algorithm with the same short lookahead makes $O(n^2k)$ evaluations, changes $O(\varepsilon^{-2})$ elements per arrival, and attains the same ratio in expectation at each fixed time. The main tool is a greedy algorithm with an explicit guarantee for each of its partial solutions combined with each initial segment of the known future arrivals. A second tool moves between two such solutions one element at a time while keeping the value high, and the deterministic algorithm times each move using upper bounds on the optimum from failed searches.

Authors: Vaneet Aggarwal

Elements arrive one at a time, and after each arrival an algorithm holds a set of at most $k$ arrived elements. Its goal is a held set whose value under a monotone submodular function is close to the best value that $k$ arrived elements can achieve. The recourse of an update is the number of elements inserted into or removed from the held set, and we require it to be bounded independently of $k$ and of the number $n$ of arrivals. With evaluations only on sets of arrived elements and polynomially many of them, the best ratio a randomized algorithm can achieve is $2-\sqrt{2}\approx0.5858$. We show that seeing future arrivals raises the best ratio for polynomially many evaluations to $1-1/e\approx0.6321$, for deterministic and for randomized algorithms; no algorithm with polynomially many evaluations exceeds $1-1/e$, even when all elements are available at once. For every $\varepsilon>0$, a deterministic algorithm that sees the next $\lceil\varepsilon k/3\rceil$ arrivals keeps a set of value at least $1-1/e-\varepsilon$ times the optimum at every time, changes $O(\varepsilon^{-3})$ elements per arrival and makes $(n+1)^{O(\varepsilon^{-2})}$ evaluations. With the whole arrival sequence known in advance, a deterministic algorithm attains the same ratio with $O(\varepsilon^{-2})$ changes per arrival. A randomized algorithm with the same short lookahead makes $O(n^2k)$ evaluations, changes $O(\varepsilon^{-2})$ elements per arrival, and attains the same ratio in expectation at each fixed time. The main tool is a greedy algorithm with an explicit guarantee for each of its partial solutions combined with each initial segment of the known future arrivals. A second tool moves between two such solutions one element at a time while keeping the value high, and the deterministic algorithm times each move using upper bounds on the optimum from failed searches.

Tuesday, November 10 2026, 01:00

Parametrize Integer Programming by Packing Imbalanced Circuits

Authors: Yuheng Shi

We study the standard form Integer Programming problem (IP) parametrized by the imbalanced circuit packing number. This number is $0$ if and only if $A$ can be transformed to a totally unimodular matrix by row operations, in which case (IP) is solvable in polynomial time. We show that if the constraint matrix represents a graphic matroid, (IP) is fixed-parameter tractable when parametrized by the imbalanced circuit packing number of the matrix. The proof combines an Erdős-Pósa property for edge-colored non-monochromatic cycles with a reduction to mixed-integer programming using few integer variables. In contrast, by a reduction from the parity potential problem, we show NP-completeness of feasibility of (IP) when the constraint matrix represents a cographic matroid, even if kernel of the constraint matrix is $2$-regular and imbalanced circuits pairwise intersect.

Authors: Yuheng Shi

We study the standard form Integer Programming problem (IP) parametrized by the imbalanced circuit packing number. This number is $0$ if and only if $A$ can be transformed to a totally unimodular matrix by row operations, in which case (IP) is solvable in polynomial time. We show that if the constraint matrix represents a graphic matroid, (IP) is fixed-parameter tractable when parametrized by the imbalanced circuit packing number of the matrix. The proof combines an Erdős-Pósa property for edge-colored non-monochromatic cycles with a reduction to mixed-integer programming using few integer variables. In contrast, by a reduction from the parity potential problem, we show NP-completeness of feasibility of (IP) when the constraint matrix represents a cographic matroid, even if kernel of the constraint matrix is $2$-regular and imbalanced circuits pairwise intersect.

Tuesday, November 10 2026, 01:00

Tunnels: Compressing the Solution Subgraph of Regular Path Queries

Authors: Adrián Gómez-Brandón, Gonzalo Navarro

Regular path queries (RPQs) are increasingly expected to return the paths that match them, not only their endpoints. Since the number of matching paths can be exponential or infinite, a standard way to deliver them (PMR) is the subgraph $\Hs$ of the product between the graph and the query automaton that contains exactly the nodes lying on some solution path. We show that $\Hs$ can be compressed much further when the graph is available to the decoder: most of $\Hs$ can be regenerated by re-running small parts of the traversal, and only the places where the traversal would waste significant effort on dead branches need to be stored. We formalize this as \emph{tunnels}: regions of $\Hs$ that are re-expanded from a stored entry point with a bounded amount of extra work, either an absolute budget or one proportional to the output produced. We handle the three classes of RPQs: constant-to-variable, constant-to-constant, and variable-to-variable. In the latter, we study the combination of tunnels with cutting the query at a selective articulation point of the automaton and encoding the answer as a union of Cartesian products of leftward and rightward traversals. Our experimental results show improvements of 2 to 3 orders of magnitude over the baseline space of PMR, for an acceptable increase in traversal time.

Authors: Adrián Gómez-Brandón, Gonzalo Navarro

Regular path queries (RPQs) are increasingly expected to return the paths that match them, not only their endpoints. Since the number of matching paths can be exponential or infinite, a standard way to deliver them (PMR) is the subgraph $\Hs$ of the product between the graph and the query automaton that contains exactly the nodes lying on some solution path. We show that $\Hs$ can be compressed much further when the graph is available to the decoder: most of $\Hs$ can be regenerated by re-running small parts of the traversal, and only the places where the traversal would waste significant effort on dead branches need to be stored. We formalize this as \emph{tunnels}: regions of $\Hs$ that are re-expanded from a stored entry point with a bounded amount of extra work, either an absolute budget or one proportional to the output produced. We handle the three classes of RPQs: constant-to-variable, constant-to-constant, and variable-to-variable. In the latter, we study the combination of tunnels with cutting the query at a selective articulation point of the automaton and encoding the answer as a union of Cartesian products of leftward and rightward traversals. Our experimental results show improvements of 2 to 3 orders of magnitude over the baseline space of PMR, for an acceptable increase in traversal time.

Tuesday, November 10 2026, 01:00

An Improved Algorithm for Quantified Boolean Formulas

Authors: Zhiyang Chen, Hailong Yao

Quantified Boolean formula (QBF), the quantified version of Boolean satisfiability (SAT), is a fundamental problem in constraint solving. Due to the $\mathsf{PSPACE}$-complete nature of QBF, designing theoretically efficient QBF algorithms is much harder and receives less attention in academia. In this work, we present a new algorithm based on memoized search, which solves QBF with $n$ variables and $m$ clauses in $1.685^m\cdot\text{poly}(n)$ time and space. This is the first improvement over the $1.709^m\cdot\text{poly}(n)$ bound by Williams in over 20 years. Our technique is based on the minimum vertex cover of the conflict graphs, which was proposed to study the parameterized complexity of SAT. We extend this idea to QBF and further show that QBF is slicewise polynomial-time solvable using this parameter.

Authors: Zhiyang Chen, Hailong Yao

Quantified Boolean formula (QBF), the quantified version of Boolean satisfiability (SAT), is a fundamental problem in constraint solving. Due to the $\mathsf{PSPACE}$-complete nature of QBF, designing theoretically efficient QBF algorithms is much harder and receives less attention in academia. In this work, we present a new algorithm based on memoized search, which solves QBF with $n$ variables and $m$ clauses in $1.685^m\cdot\text{poly}(n)$ time and space. This is the first improvement over the $1.709^m\cdot\text{poly}(n)$ bound by Williams in over 20 years. Our technique is based on the minimum vertex cover of the conflict graphs, which was proposed to study the parameterized complexity of SAT. We extend this idea to QBF and further show that QBF is slicewise polynomial-time solvable using this parameter.

Tuesday, November 10 2026, 01:00

A Simple Computer-Assisted Proof of the 5/6-Bound for Real-Period Pinwheel Scheduling

Authors: Hiroshi Fujiwara, Grzegorz Gutowski, Katsuhisa Ouchi, Alexander Wolff

In pinwheel scheduling with real periods, an instance consists of $n$ tasks with real periods $1\le a_1\le a_2\le\cdots\le a_n$. The goal is to find a schedule that assigns one task to each day indefinitely such that, for every task $i\in\{1,\dots,n\}$ and every positive integer $\ell$, task $i$ occurs at least $\ell$ times in every block of $\lceil\ell a_i\rceil$ consecutive days. Given an instance $(a_1,a_2,\dots,a_n)$, its dimension is $n$, its density is $\sum_{i=1}^n1/a_i$, and its potential is $\sum_{i=1}^{n-1}1/a_i$. We prove that every real-valued instance of density at most 5/6 admits a periodic schedule of period length less than $4\cdot2^n$. For integer periods, the 5/6 density bound was a long-standing conjecture settled by Kawamura using a computer-assisted proof [STOC 2024, PNAS 2026]. Our result confirms the real-period conjecture of Fujiwara, Miyagi, and Ouchi [DMTCS 2026]. Our computer-assisted proof builds on Kawamura's ideas while using fewer explicit schedules. It combines dimension reduction by folding with nine satisfiability modulo theories (SMT) checks over linear real arithmetic. A set of 14287 explicit periodic schedules contains a schedule for every instance of dimension at most 8 and density at most 5/6, and for every instance of dimension 9 and potential at most 5/6. Folding is used to reduce every instance of larger dimension and potential at most 5/6 to an instance of dimension 9 and the same potential bound. We also prove that every real-valued instance with minimum period at least 4 and density at most 17/20=0.85 admits a periodic schedule of period length less than $19\cdot2^n$. The density bound 17/20 is tight and improves the 0.84-bound of Mishra, Rho, and Kleinberg [arXiv 2025] for integer periods. It also resolves the $a_1=4$ case of Kawamura's 2025 conjecture that integer-valued instances of density at most $1-(a_1-1)/(a_1(a_1+1))$ are schedulable.

Authors: Hiroshi Fujiwara, Grzegorz Gutowski, Katsuhisa Ouchi, Alexander Wolff

In pinwheel scheduling with real periods, an instance consists of $n$ tasks with real periods $1\le a_1\le a_2\le\cdots\le a_n$. The goal is to find a schedule that assigns one task to each day indefinitely such that, for every task $i\in\{1,\dots,n\}$ and every positive integer $\ell$, task $i$ occurs at least $\ell$ times in every block of $\lceil\ell a_i\rceil$ consecutive days. Given an instance $(a_1,a_2,\dots,a_n)$, its dimension is $n$, its density is $\sum_{i=1}^n1/a_i$, and its potential is $\sum_{i=1}^{n-1}1/a_i$. We prove that every real-valued instance of density at most 5/6 admits a periodic schedule of period length less than $4\cdot2^n$. For integer periods, the 5/6 density bound was a long-standing conjecture settled by Kawamura using a computer-assisted proof [STOC 2024, PNAS 2026]. Our result confirms the real-period conjecture of Fujiwara, Miyagi, and Ouchi [DMTCS 2026]. Our computer-assisted proof builds on Kawamura's ideas while using fewer explicit schedules. It combines dimension reduction by folding with nine satisfiability modulo theories (SMT) checks over linear real arithmetic. A set of 14287 explicit periodic schedules contains a schedule for every instance of dimension at most 8 and density at most 5/6, and for every instance of dimension 9 and potential at most 5/6. Folding is used to reduce every instance of larger dimension and potential at most 5/6 to an instance of dimension 9 and the same potential bound. We also prove that every real-valued instance with minimum period at least 4 and density at most 17/20=0.85 admits a periodic schedule of period length less than $19\cdot2^n$. The density bound 17/20 is tight and improves the 0.84-bound of Mishra, Rho, and Kleinberg [arXiv 2025] for integer periods. It also resolves the $a_1=4$ case of Kawamura's 2025 conjecture that integer-valued instances of density at most $1-(a_1-1)/(a_1(a_1+1))$ are schedulable.

Tuesday, November 10 2026, 01:00

Close and Sort: Exact Maximum-Likelihood Aggregation of Cyclic Rank-Two Preferences

Authors: Kehan Liu, Zhaoyun Chen, Xianquan Yan

Pairwise probabilities arise in rank aggregation, recommendation, and human or LLM evaluation, yet converting them into a maximum-likelihood list is a weighted Kemeny problem and is NP-hard in general. Known exact algorithms for rank-two tournaments use only comparison signs and do not optimize this likelihood. We show that the weighted problem is exactly solvable when the skew-symmetric log-odds matrix $W_{ij}=\log(p_{ij}/p_{ji})$ has rank two, including when the induced preferences contain cycles. Writing $W_{ij}=\det(h_i,h_j)$ and adding the closing vector $h_0=-\sum_i h_i$ turns the objective into twice the signed area of a closed polygon. This geometry yields CloseSort, which polar-sorts the $n+1$ edge vectors, cuts after $h_0$, and returns a global maximum-likelihood order in $O(n\log n)$ time from the factors. The same geometry identifies the runner-up and gives a stability certificate beyond exact rank two. When the smallest determinant $δ$ of two consecutive edges in the closed angular cycle is positive, it is exactly the best-to-runner-up log-likelihood gap. For $W=W_0+E$, the CloseSort order of $W_0$ has regret at most $\sum_{i

Authors: Kehan Liu, Zhaoyun Chen, Xianquan Yan

Pairwise probabilities arise in rank aggregation, recommendation, and human or LLM evaluation, yet converting them into a maximum-likelihood list is a weighted Kemeny problem and is NP-hard in general. Known exact algorithms for rank-two tournaments use only comparison signs and do not optimize this likelihood. We show that the weighted problem is exactly solvable when the skew-symmetric log-odds matrix $W_{ij}=\log(p_{ij}/p_{ji})$ has rank two, including when the induced preferences contain cycles. Writing $W_{ij}=\det(h_i,h_j)$ and adding the closing vector $h_0=-\sum_i h_i$ turns the objective into twice the signed area of a closed polygon. This geometry yields CloseSort, which polar-sorts the $n+1$ edge vectors, cuts after $h_0$, and returns a global maximum-likelihood order in $O(n\log n)$ time from the factors. The same geometry identifies the runner-up and gives a stability certificate beyond exact rank two. When the smallest determinant $δ$ of two consecutive edges in the closed angular cycle is positive, it is exactly the best-to-runner-up log-likelihood gap. For $W=W_0+E$, the CloseSort order of $W_0$ has regret at most $\sum_{i

Tuesday, November 10 2026, 01:00

Combinatorial optimization with bounded subdeterminants

Authors: Samuel Fiorini, J. Pascal Gollin, Tony Huynh, Simon Keil, O-joung Kwon, Sang-il Oum, Roohani Sharma, Stefan Weltge

We take a first step toward a broader study of classical combinatorial optimization problems through the maximum subdeterminants of their standard integer programming formulations, yielding new positive results for several famous (NP-hard) problems. An integer matrix is totally $Δ$-modular if every square submatrix has determinant of absolute value at most $Δ$. For every fixed $Δ$, it has been known that maximum clique, maximum stable set, and minimum vertex cover can be solved in polynomial time on instances for which the coefficient matrix of the natural formulation is totally $Δ$-modular. We extend this list by showing that the same holds for many other problems, including the weighted versions of maximum cut, the traveling salesman problem, feedback vertex set, odd cycle transversal, dominating set, max 2-sat, induced matching, and cluster vertex deletion, as well as for $k$-colorability and graph isomorphism. Our proofs reveal several new ways in which bounded subdeterminants impose combinatorial structure: they bound the size or maximum degree of the input graph, bound packings of cycles and other obstructions, or yield decompositions into tractable graph classes such as forests, chordal graphs, graphs of bounded treewidth, and graphs of bounded linear mim-width.

Authors: Samuel Fiorini, J. Pascal Gollin, Tony Huynh, Simon Keil, O-joung Kwon, Sang-il Oum, Roohani Sharma, Stefan Weltge

We take a first step toward a broader study of classical combinatorial optimization problems through the maximum subdeterminants of their standard integer programming formulations, yielding new positive results for several famous (NP-hard) problems. An integer matrix is totally $Δ$-modular if every square submatrix has determinant of absolute value at most $Δ$. For every fixed $Δ$, it has been known that maximum clique, maximum stable set, and minimum vertex cover can be solved in polynomial time on instances for which the coefficient matrix of the natural formulation is totally $Δ$-modular. We extend this list by showing that the same holds for many other problems, including the weighted versions of maximum cut, the traveling salesman problem, feedback vertex set, odd cycle transversal, dominating set, max 2-sat, induced matching, and cluster vertex deletion, as well as for $k$-colorability and graph isomorphism. Our proofs reveal several new ways in which bounded subdeterminants impose combinatorial structure: they bound the size or maximum degree of the input graph, bound packings of cycles and other obstructions, or yield decompositions into tractable graph classes such as forests, chordal graphs, graphs of bounded treewidth, and graphs of bounded linear mim-width.

Tuesday, November 10 2026, 01:00

The Left Ripple of Byte-Pair Encoding: Certified Lookbehind for Every Dictionary

Authors: Nicklas Nidhögg

Berglund and van der Merwe formalized byte-pair encoding and bounded how far an appended string can change a tokenization, leaving the converse open: how far can a string prepended on the left change it, which dictionaries make that ripple infinite, and can it be computed efficiently when finite? We answer all three for every dictionary. The origin radius, how far a prepend can change covering spans, is decided with radius certificates by a product of two runs of the automaton that guesses the final tokens, in time polynomial in the dictionary written out, and it is infinite exactly when one periodic word has two canonical tilings with no common boundary. Under their ordering condition it is infinite exactly when some rule (t, t) finds its two copies adjacent at its own stage, and such a rule makes any dictionary infinite. A finite radius makes every window of a width read off the radii have a cut, and the radius is finite exactly when such cuts are dense. The token machine completes for every dictionary, and the tokenization is subsequential, extending what Berglund, Martens and van der Merwe proved for proper ones under one pass per rule. A streaming tokenizer emits after every symbol exactly the tokens that no continuation changes, in polynomial time, answering a question of Dalirrooyfard and Ye. Boundedness of the label measure, the edit distance of the token sequences, is decided for every dictionary in polynomial time as well, and under the ordering condition the largest distance one edit causes over dictionaries of m rules lies asymptotically between m^2/10 and m^2/2. GPT-2's merge table carries 244 effective self-pairs, so its origin radius is infinite from the forty-second merge on, and that of cl100k, o200k and Llama 3 from their first. Every measured number is its program's output, and what is not executed is named.

Authors: Nicklas Nidhögg

Berglund and van der Merwe formalized byte-pair encoding and bounded how far an appended string can change a tokenization, leaving the converse open: how far can a string prepended on the left change it, which dictionaries make that ripple infinite, and can it be computed efficiently when finite? We answer all three for every dictionary. The origin radius, how far a prepend can change covering spans, is decided with radius certificates by a product of two runs of the automaton that guesses the final tokens, in time polynomial in the dictionary written out, and it is infinite exactly when one periodic word has two canonical tilings with no common boundary. Under their ordering condition it is infinite exactly when some rule (t, t) finds its two copies adjacent at its own stage, and such a rule makes any dictionary infinite. A finite radius makes every window of a width read off the radii have a cut, and the radius is finite exactly when such cuts are dense. The token machine completes for every dictionary, and the tokenization is subsequential, extending what Berglund, Martens and van der Merwe proved for proper ones under one pass per rule. A streaming tokenizer emits after every symbol exactly the tokens that no continuation changes, in polynomial time, answering a question of Dalirrooyfard and Ye. Boundedness of the label measure, the edit distance of the token sequences, is decided for every dictionary in polynomial time as well, and under the ordering condition the largest distance one edit causes over dictionaries of m rules lies asymptotically between m^2/10 and m^2/2. GPT-2's merge table carries 244 effective self-pairs, so its origin radius is infinite from the forty-second merge on, and that of cl100k, o200k and Llama 3 from their first. Every measured number is its program's output, and what is not executed is named.

Tuesday, November 10 2026, 01:00

TR26-315 | On derandomizing the line-point low-degree test | Prahladh Harsha, Mrinal Kumar, Madhu Sudan, Swastik Kopparty, Ramprasad Saptharishi

from ECCC Papers

We investigate the derandomized version of the classical line-point low-degree test in the high-error regime in the bivariate setting and reveal some unexpected behavior. The classical line-point low-degree test takes an $m$-variate function $f$ over a finite field $\mathbb{F}_q$, queries it along a random line and accepts if the restriction of the function to this line is a low-degree polynomial. The high-error version of (the analysis of) this test asserts that if $f$ has at least $\epsilon$ expected local agreement with a degree-$k$ univariate polynomial on a random line, then in fact $f$ is non-trivially close to some degree-$k$ $m$-variate polynomial even as $\epsilon$ approaches $0$ (provided $q$ is sufficiently large compared to $k$ and $1/\epsilon$). The derandomized version of the problem asks if the same holds when the line is not chosen uniformly over all possible lines but rather from a small set. Derandomized analyses of low-degree tests play a crucial role in obtaining near-linear sized PCPs. In this paper we explore the bivariate case ($m=2$). We show, somewhat surprisingly, that there exists a function $f\colon \mathbb{F}_q^2 \to \mathbb{F}_q$ and a set of lines $\mathcal{L}$ in $\mathbb{F}_q^2$ of size nearly $q^{3/2}$ such that on every line in $\mathcal{L}$, $f$ has agreement $\approx 1/2$ on average with some univariate polynomial of degree $k=\sqrt{q}/2$ and yet $f$ is $1 - O(q^{-1/2})$ far from every polynomial of degree $O(k)$. On the positive side, we show that once the field size $q$ is at least $k^2/\text{poly}(\epsilon)$ such examples disappear and we can get sets of lines of size $O(q/\epsilon^{O(1)})$ on which the low-degree test has a high-error analysis. On the way to our positive result, we show a new high-degree version of the classical two-direction Polishchuk--Spielman divisibility lemma. It deals with finding a low-degree global root $Z = U(X,Y)$ of a trivariate polynomial $Q(X,Y,Z)$, based on restrictions of $Q$ to lines in $r > \deg_Z(Q)$ different directions (generalizing the original $r = 2$ and $\deg_Z(Q) = 1$ case). Specifically, it states that if $Q$ has a low-degree root $Z = U_{a,b}(X)$ for all lines $Y = aX + b$ in each of those $r$ directions, then $Q$ has a low-degree global root $Z = U(X,Y)$. This high degree Polishchuk--Spielman lemma is proved by a new technique: using simultaneous Hensel lifting of "consistent" roots on many parallel lines. Our negative result addresses one of the big challenges in analyzing low-degree tests, namely the rarity of interesting counterexamples that offer concrete barriers.

Monday, November 09 2026, 21:32

TR26-314 | Polynomial-size hitting sets for multilinear any-order read-once algebraic branching programs and for set-multilinear depth-$3$ circuits with constant-size variable sets | Amir Shpilka

from ECCC Papers

In this paper we give polynomial-size hitting sets for multilinear any-order read-once oblivious algebraic branching programs and for depth-$3$ set-multilinear circuits where each variable set has constant size. Our results also extend to the border of both classes and hold over fields of any characteristic. The proof combines the approach of Forbes, Saptharishi and Shpilka (STOC 2014) with the recent results of Bhardwaj and Saptharishi (2026) and of Bal and Ghosh (2026) that gave polynomial-size hitting sets for polynomials with small partial derivative dimension.

Monday, November 09 2026, 15:32

TR26-313 | A Little Derandomization Goes a Long Way: Unconditional Circuit Lower Bounds using Low-Degree PIT | Nikolai Chukhin, Karthik Gajulapalli, Alexander Golovnev, Ivan Mihajlin

from ECCC Papers

We prove unconditionally that either NE contains a language requiring Boolean circuits of size at least $2^n/(2n)$, or $VP^0_{\mathbb{Z}}\ne VNP^0_{\mathbb{Z}}$, a constant-free version of Valiant's conjecture over the integers. The Boolean alternative implies that $NP\not\subseteq SIZE[n^k]$ for every constant $k\ge1$. Our proof gives a fine-grained variant of the connection between Polynomial Identity Testing (PIT) and circuit lower bounds established by Kabanets and Impagliazzo (2004). We show that the specific identity tests required by this argument can be performed over the integers without a derandomization assumption. Over a fixed finite field $\mathbb{F}$, we first give a deterministic $O(st^{d-1})$-time algorithm for PIT on size-$s$ circuits computing degree-$d$ polynomials in $t$ variables. We further show that, for any fixed $d\ge2$ and $\varepsilon>0$, improving this running time by a factor of $t^{\varepsilon}$, even nondeterministically, implies either the same Boolean lower bound or an arithmetic circuit lower bound of $t^{1+\varepsilon/2}$ for a polynomial family from $VNP_{\mathbb{F}}$. If nondeterministic PIT instead runs in time $s^{o(d)}t^{(1-\varepsilon)d}$ for infinitely many degrees $d$, then the arithmetic lower bound strengthens to $VP_{\mathbb{F}}\ne VNP_{\mathbb{F}}$. Over the integers, we give a deterministic PIT algorithm for syntactically set-multilinear circuits running in time $s^{O(1)}t^{5d/6}$, which yields the unconditional result above. Combining the Kabanets–Impagliazzo argument with range avoidance, we show that an NP algorithm for PIT over $\mathbb{Z}$ gives a purely arithmetic win–win: either $VP_{\mathbb{Z}}\ne VNP_{\mathbb{Z}}$, or constant-degree polynomials of maximum circuit complexity can be constructed in FNP. Along the way, we show that arithmetic circuits of size $s(N)$ for the permanent, or for an explicit family in VNP, give perfectly complete input-oblivious Merlin–Arthur protocols running in time $s(O(n))\,n^{O(1)}$ for 3-UNSAT with $O(n)$ clauses; thus ruling out faster protocols yields arithmetic lower bounds that scale with the protocol's running time.

Monday, November 09 2026, 14:01

Ph.D. student at West Virginia University (apply by December 15, 2026)

from CCI: jobs

My lab has a fully funded position for a Ph.D. student. Skills needed: Linear and Integer Programming, some exposure to programming. Undergrad in CS or Industrial Engineering. Website: community.wvu.edu/~krsubramani/ Email: k.subramani@mail.wvu.edu

My lab has a fully funded position for a Ph.D. student. Skills needed: Linear and Integer Programming, some exposure to programming. Undergrad in CS or Industrial Engineering.

Website: https://community.wvu.edu/~krsubramani/
Email: k.subramani@mail.wvu.edu

By shacharlovett

Monday, November 09 2026, 12:26

Demaine-Gasarch-Hajiaghayi have written a book that is a sequel to Garey and Johnson

from Computational Complexity

LANCE: You should take a break from all the AI stuff and talk about your book!

BILL: I will! But it's not my book its co-authored with Erik Demaine and Mohammad Hajiaghayi. The title is

Computational Intractability: A Guide to Algorithmic Lower Bounds

LANCE: That sounds similar to Garey and Johnson's classic

Computers and Intractability: A Guide to the Theory of NP-Completeness

BILL: Indeed. DGH is a sequel to Garey and Johnson.

LANCE: But so much has happened since 1979.

BILL: Yes. That's why the book is over 500 pages and still does not cover everything since 1979.

LANCE: So what IS in the book?

BILL: The main question that drives complexity theory (and many other fields) is the question:

Given a problem, how hard is it?

LANCE: Oh, You mean like P and NP!

BILL: Yes, but there is more! We cover P and NP: Many problems are proven NP-complete. Some are on Puzzles and Games.

Examples of NP-Complete Problems that you might not have heard of

Disk Packing: Given a set of disks and a target square, can you place the disks so that they do not overlap and all of their centers lie within the square? This is Strongly NP-hard.

Mate-in-1: Given a position in Settlers of Catan, can the player who is about to go win? This is NP-complete

Losing Checkers: Given a position in Checkers, is there a move that forces your opponent to win? This is NP-complete. This is an important problem in case you play Checkers with a Wookie, see here.

Super Mario Brothers and Many other Nintendo Games: Given a position in the game, and a level, can the player get to that level? This is NP-complete. (There are versions that are PSPACE-complete and versions that are undecidable)

Some use the machinery developed since G & J to show lower bounds on approximation:

VC with vertices of degree $\le 3$: If there is a polytime $(\frac{145}{144}-\epsilon)$-approx then P=NP.

We have a chapter on the Unique Games Conj--uh, uh, Theorem.

We look BELOW NP: 3SUM-hardness, APSP-hardness (these are both a bit outdated), streaming algorithms, online algorithms, others.

We look ABOVE NP: #P, ETR,

LANCE: What's ETR?

BILL: Existential Theory of the Reals is the following problem: Given a sentence of the form

$(\exists x_1)(\exists x_2)\cdots(\exists x_n)[ C_1 \wedge \cdots \wedge C_k]$

where each $C_i$ is a polynomial equality or inequality in $x_1,\ldots,x_n$

is this sentence, with the quantifiers going over the reals, true?

Tarski proved ETR is decidable. ETR is known to be NP-hard and is thought to be harder than NP.

A problem is ETR-complete if it is equivalent to ETR. This is a way to classify some problems that involve reals that were hard to classify with more traditional classes. For example, the Art Gallery Problem (how many guards do you need) is ETR-complete

LANCE: What else do you have that's above NP?

BILL: Poly Hierarchy, PSPACE, and even undecidability.

LANCE: Are there really NEW results about undecidability?

BILL: There are some video games where questions about winning are undecidable. I doubt Turing knew that. One of them is called braid (see here).

LANCE: Don't you have whole chapters devoted to the now-defunct Exponential Time Hypothesis and the 3-SUM conjecture?

Bill: Also the APSP conjecture. Are these now-defunct conjectures still worth studying? I did a blog post on that here. Do you read my blog posts?

LANCE: Do you read mine?

BILL: Fair point.

LANCE: Despite having three chapters obsolete, the rest of the book sounds great. Where can I get it?

BILL: You can buy it on Amazon, see here. You can buy it at MIT press here. You can download a prior draft here.

By gasarch

LANCE: You should take a break from all the AI stuff and talk about your book!

BILL: I will! But it's not my book its co-authored with Erik Demaine and Mohammad Hajiaghayi. The title is

Computational Intractability: A Guide to Algorithmic Lower Bounds

LANCE: That sounds similar to Garey and Johnson's classic

Computers and Intractability: A Guide to the Theory of NP-Completeness

BILL: Indeed. DGH is a sequel to Garey and Johnson.

LANCE: But so much has happened since 1979.

BILL: Yes. That's why the book is over 500 pages and still does not cover everything since 1979.

LANCE: So what IS in the book?

BILL: The main question that drives complexity theory (and many other fields) is the question:

Given a problem, how hard is it?

LANCE: Oh, You mean like P and NP!

BILL: Yes, but there is more! We cover P and NP: Many problems are proven NP-complete. Some are on Puzzles and Games.

Examples of NP-Complete Problems that you might not have heard of

Disk Packing: Given a set of disks and a target square, can you place the disks so that they do not overlap and all of their centers lie within the square? This is Strongly NP-hard.

Mate-in-1: Given a position in Settlers of Catan, can the player who is about to go win? This is NP-complete

Losing Checkers: Given a position in Checkers, is there a move that forces your opponent to win? This is NP-complete. This is an important problem in case you play Checkers with a Wookie, see here.

Super Mario Brothers and Many other Nintendo Games: Given a position in the game, and a level, can the player get to that level? This is NP-complete. (There are versions that are PSPACE-complete and versions that are undecidable)

Some use the machinery developed since G & J to show lower bounds on approximation:

VC with vertices of degree $\le 3$: If there is a polytime $(\frac{145}{144}-\epsilon)$-approx then P=NP.

We have a chapter on the Unique Games Conj--uh, uh, Theorem.

We look BELOW NP: 3SUM-hardness, APSP-hardness (these are both a bit outdated), streaming algorithms, online algorithms, others.

We look ABOVE NP: #P, ETR,

LANCE: What's ETR?

BILL: Existential Theory of the Reals is the following problem: Given a sentence of the form

$(\exists x_1)(\exists x_2)\cdots(\exists x_n)[ C_1 \wedge \cdots \wedge C_k]$

where each $C_i$ is a polynomial equality or inequality in $x_1,\ldots,x_n$

is this sentence, with the quantifiers going over the reals, true?

Tarski proved ETR is decidable. ETR is known to be NP-hard and is thought to be harder than NP.

A problem is ETR-complete if it is equivalent to ETR. This is a way to classify some problems that involve reals that were hard to classify with more traditional classes. For example, the Art Gallery Problem (how many guards do you need) is ETR-complete

LANCE: What else do you have that's above NP?

BILL: Poly Hierarchy, PSPACE, and even undecidability.

LANCE: Are there really NEW results about undecidability?

BILL: There are some video games where questions about winning are undecidable. I doubt Turing knew that. One of them is called braid (see here).

LANCE: Don't you have whole chapters devoted to the now-defunct Exponential Time Hypothesis and the 3-SUM conjecture?

Bill: Also the APSP conjecture. Are these now-defunct conjectures still worth studying? I did a blog post on that here. Do you read my blog posts?

LANCE: Do you read mine?

BILL: Fair point.

LANCE: Despite having three chapters obsolete, the rest of the book sounds great. Where can I get it?

BILL: You can buy it on Amazon, see here. You can buy it at MIT press here. You can download a prior draft here.

By gasarch

Monday, November 09 2026, 01:31

Beyond the $n/2$ Threshold: Quantum Query Complexity for Computing Hamming Weight and Symmetric Functions

Authors: Yuan Deng, Zihan Hao, Honghao Lin, Qipeng Liu, Vahab Mirrokni, Jianan Shao, David P. Woodruff

The oracle-interrogation algorithm [van Dam, FOCS '98] computes every $n$-bit Boolean function with bounded error using $n/2+O(\sqrt n)$ quantum queries. In contrast, parity can be computed exactly using $\lceil n/2\rceil$ queries, but whether an additive overhead beyond $n/2$ is necessary for a single Boolean output has remained open. In this work, we construct an explicit family of symmetric Boolean functions requiring $n/2+Ω(n^{1/3})$ quantum queries at a fixed positive constant error. We also give an algorithm that recovers the exact Hamming weight using $n/2+O(n^{1/3})$ quantum queries with bounded error, and therefore computes every symmetric Boolean function within the same query bound. Together, these results determine the optimal additive order for Hamming weight recovery and for the worst case over symmetric Boolean functions at sufficiently small constant error.

Authors: Yuan Deng, Zihan Hao, Honghao Lin, Qipeng Liu, Vahab Mirrokni, Jianan Shao, David P. Woodruff

The oracle-interrogation algorithm [van Dam, FOCS '98] computes every $n$-bit Boolean function with bounded error using $n/2+O(\sqrt n)$ quantum queries. In contrast, parity can be computed exactly using $\lceil n/2\rceil$ queries, but whether an additive overhead beyond $n/2$ is necessary for a single Boolean output has remained open. In this work, we construct an explicit family of symmetric Boolean functions requiring $n/2+Ω(n^{1/3})$ quantum queries at a fixed positive constant error. We also give an algorithm that recovers the exact Hamming weight using $n/2+O(n^{1/3})$ quantum queries with bounded error, and therefore computes every symmetric Boolean function within the same query bound. Together, these results determine the optimal additive order for Hamming weight recovery and for the worst case over symmetric Boolean functions at sufficiently small constant error.

Monday, November 09 2026, 01:00

A proof of the Aaronson-Ambainis conjecture