Theory of Computing Report

Authors: Yuan Huang, Zhiguo Fu

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

Authors: Yuan Huang, Zhiguo Fu

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

Friday, September 11 2026, 00:00

Online Treasure Hunt in Vertex-Permuted Dynamic Rings

Authors: Kamran Ayoubi, Bernard Mans, Lata Narayanan

We study the problem of treasure hunt by a group of $k \geq 1$ agents in vertex-permuted dynamic rings (VP). In this model, the $n$ vertices remain on a ring but are permuted at each time step. We first show that treasure hunt is impossible for any $k \leq n-3$ agents, if there are no restrictions on the sequence of permutations used in the dynamic ring. We then study the $VP(δ)$ setting, in which for every pair $i, j$ of vertices, the edge $(i, j)$ is guaranteed to appear within $δ$ steps. We show that the class $VP(δ)$ is feasible only for $δ\geq \left\lceil \frac{n-1}{2}\right\rceil$. For the one-agent case, we show a tight bound of $Θ(δn)$ on the worst-case search time as well as competitive ratio of any online algorithm for treasure hunt, provided $δ\geq 2n$. We then give an optimal algorithm for $k$ agents, thereby showing that $k$ agents can obtain a speedup of $k$ on the worst-case search time. Finally, in the R-VP setting, in which in every step, the vertices are arranged as a ring according to a random permutation, we show that treasure hunt takes expected $Θ(n)$ steps against an oblivious adversary and $Θ(n \log n)$ steps against an adaptive adversary.

Authors: Kamran Ayoubi, Bernard Mans, Lata Narayanan

We study the problem of treasure hunt by a group of $k \geq 1$ agents in vertex-permuted dynamic rings (VP). In this model, the $n$ vertices remain on a ring but are permuted at each time step. We first show that treasure hunt is impossible for any $k \leq n-3$ agents, if there are no restrictions on the sequence of permutations used in the dynamic ring. We then study the $VP(δ)$ setting, in which for every pair $i, j$ of vertices, the edge $(i, j)$ is guaranteed to appear within $δ$ steps. We show that the class $VP(δ)$ is feasible only for $δ\geq \left\lceil \frac{n-1}{2}\right\rceil$. For the one-agent case, we show a tight bound of $Θ(δn)$ on the worst-case search time as well as competitive ratio of any online algorithm for treasure hunt, provided $δ\geq 2n$. We then give an optimal algorithm for $k$ agents, thereby showing that $k$ agents can obtain a speedup of $k$ on the worst-case search time. Finally, in the R-VP setting, in which in every step, the vertices are arranged as a ring according to a random permutation, we show that treasure hunt takes expected $Θ(n)$ steps against an oblivious adversary and $Θ(n \log n)$ steps against an adaptive adversary.

Friday, September 11 2026, 00:00

The Quantum Overlap Gap Property and Algorithmic Hardness for the Quantum Hypergraph Max-Cut Problem

Authors: Mikhail Mints, Eric R. Anschuetz

In this work, we analyze the average-case hardness of approximation for the Quantum Hypergraph Max-Cut problem using the theoretical framework of the Quantum Overlap Gap Property (QOGP). We establish two main results. Our first result applies to a wide class of stable quantum algorithms, satisfying a Lipschitz property with respect to the quantum Wasserstein distance of order $2$. We show a weak hardness result, demonstrating that for any Lipschitz constant $L$, there is some $k$ such that $L$-stable algorithms cannot approximate the optimal solution to Quantum Hypergraph Max-Cut on $k$-uniform hypergraphs in the average case. Additionally, we establish a strong hardness result where $k$ is independent of $L$, but only for a more restricted class of local quantum algorithms defined using the quantum Wasserstein distance of order $\infty$. We apply these results to establish concrete depth lower bounds for popular quantum algorithms for preparing near-optimal states for this problem.

Authors: Mikhail Mints, Eric R. Anschuetz

In this work, we analyze the average-case hardness of approximation for the Quantum Hypergraph Max-Cut problem using the theoretical framework of the Quantum Overlap Gap Property (QOGP). We establish two main results. Our first result applies to a wide class of stable quantum algorithms, satisfying a Lipschitz property with respect to the quantum Wasserstein distance of order $2$. We show a weak hardness result, demonstrating that for any Lipschitz constant $L$, there is some $k$ such that $L$-stable algorithms cannot approximate the optimal solution to Quantum Hypergraph Max-Cut on $k$-uniform hypergraphs in the average case. Additionally, we establish a strong hardness result where $k$ is independent of $L$, but only for a more restricted class of local quantum algorithms defined using the quantum Wasserstein distance of order $\infty$. We apply these results to establish concrete depth lower bounds for popular quantum algorithms for preparing near-optimal states for this problem.

Friday, September 11 2026, 00:00

PureSuperQMA(exp) = BellPureSymQMA(poly) = QMA via Dimension-Free Bosonic Argmax

Authors: William Gay, Fernando Granha Jeronimo, Lenny Liu, Itai Leigh, Pei Wu, Haochen Xu

Pure-state consistency problems naturally lead to quantum proof systems in which a single pure witness must satisfy many acceptance constraints. The corresponding class $\mathsf{PureSuperQMA}$ was previously known to lie between $\mathsf{QMA}$ and $\mathsf{QMA}(2)$, and Kamminga and Rudolph (ITCS'26) conjectured that both containments are strict. In this paper, we prove the following surprising complexity collapses $$ \mathsf{QMA} = \mathsf{PureSuperQMA} = \mathsf{PureSuperQMA}(\text{exp}) = \mathsf{BellPureSymQMA}(\text{poly}) $$ Here $\mathsf{PureSuperQMA}(\text{exp})$ allows exponentially many checks which are uniformly indexed and efficiently generated, while requiring an inverse-polynomial violation margin and an inverse-polynomial fraction of violated checks for the NO cases. $\mathsf{BellPureSymQMA}(\text{poly})$ is a related model that requires the prover to give the verifier polynomially many copies of a pure state, which the verifier measures separately with logarithmic output length for each local measurement, before processing the outcomes jointly. The main technical ingredient is a dimension-free stability bound for symmetric tensor states. Our simulations use polynomially many witness registers and combine a random-pair SWAP test with a permutation-invariant lift of the original verification procedure. The key step is to show that, on the symmetric subspace, the extremal verification value is close to that of some tensor-power witness with dimension-independent error. Applying this argument to the two verification models yields both simulations. As a consequence, exact $k$-local pure-state consistency is $\mathsf{QMA}$-complete for every fixed $k\ge2$, and so are the corresponding exact bosonic and fermionic pure $N$-representability problems.

Authors: William Gay, Fernando Granha Jeronimo, Lenny Liu, Itai Leigh, Pei Wu, Haochen Xu

Pure-state consistency problems naturally lead to quantum proof systems in which a single pure witness must satisfy many acceptance constraints. The corresponding class $\mathsf{PureSuperQMA}$ was previously known to lie between $\mathsf{QMA}$ and $\mathsf{QMA}(2)$, and Kamminga and Rudolph (ITCS'26) conjectured that both containments are strict. In this paper, we prove the following surprising complexity collapses $$ \mathsf{QMA} = \mathsf{PureSuperQMA} = \mathsf{PureSuperQMA}(\text{exp}) = \mathsf{BellPureSymQMA}(\text{poly}) $$ Here $\mathsf{PureSuperQMA}(\text{exp})$ allows exponentially many checks which are uniformly indexed and efficiently generated, while requiring an inverse-polynomial violation margin and an inverse-polynomial fraction of violated checks for the NO cases. $\mathsf{BellPureSymQMA}(\text{poly})$ is a related model that requires the prover to give the verifier polynomially many copies of a pure state, which the verifier measures separately with logarithmic output length for each local measurement, before processing the outcomes jointly. The main technical ingredient is a dimension-free stability bound for symmetric tensor states. Our simulations use polynomially many witness registers and combine a random-pair SWAP test with a permutation-invariant lift of the original verification procedure. The key step is to show that, on the symmetric subspace, the extremal verification value is close to that of some tensor-power witness with dimension-independent error. Applying this argument to the two verification models yields both simulations. As a consequence, exact $k$-local pure-state consistency is $\mathsf{QMA}$-complete for every fixed $k\ge2$, and so are the corresponding exact bosonic and fermionic pure $N$-representability problems.

Friday, September 11 2026, 00:00

Oracle Separations in the Fourier Hierarchy

Authors: Atul Mantri

The Fourier hierarchy $\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdots$, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly $\mathrm{BPP}$, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level $k\ge2$. We prove that for every constant $k\ge2$ there is an oracle relative to which $\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}$. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level $k$ it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from $\mathrm{BQP}$ relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with $k$ layers is reproduced exactly by an algorithm making only $k-1$ rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.

Authors: Atul Mantri

The Fourier hierarchy $\mathrm{FH}_0\subseteq\mathrm{FH}_1\subseteq\mathrm{FH}_2\subseteq\cdots$, introduced by Shi (TCS 2005), measures a quantum computation by the number of Hadamard layers it uses. Between two layers the circuit may permute basis states and attach phases, but it may not create superposition; the layers are its only source of interference. The first level is exactly $\mathrm{BPP}$, while the second already solves Simon's problem and, through phase estimation, factors integers. Shi conjectured that every additional layer strictly increases computational power, and asked, as a first step, for oracle separations between consecutive levels. To our knowledge, the question was open at every level $k\ge2$. We prove that for every constant $k\ge2$ there is an oracle relative to which $\mathrm{FH}_k\subsetneq\mathrm{FH}_{k+1}$. The separating problem is built from Forrelation (Aaronson and Ambainis, STOC 2015): the level above solves it with a constant number of queries, whereas at level $k$ it stays hard even for circuits making exponentially many queries. This holds for both of the usual ways of giving a circuit access to an oracle, the phase oracle and the standard oracle, which writes its answer into a register. The two are not interchangeable: relative to an oracle, the standard oracle is strictly more powerful at the same number of layers. We also separate the union of all the levels from $\mathrm{BQP}$ relative to an oracle. The lower bounds rest on a structural property of the hierarchy: the number of Hadamard layers limits how adaptively a circuit can query its oracle. With a phase oracle, a circuit with $k$ layers is reproduced exactly by an algorithm making only $k-1$ rounds of parallel queries, which brings known lower bounds for such algorithms to bear. The standard oracle lets a circuit branch on earlier answers, and that case needs a separate argument.

Friday, September 11 2026, 00:00

Topology inside NC$^1$

Authors: Eric Allender, Samir Datta, Arsenii Karnaukhov, Sambuddha Roy, Alexander Shekhovstov

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

Authors: Eric Allender, Samir Datta, Arsenii Karnaukhov, Sambuddha Roy, Alexander Shekhovstov

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

Friday, September 11 2026, 00:00

Some results on Archdeacon's conjecture for rotation systems

Authors: Arahat Chikkatur, Ji Zeng

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

Authors: Arahat Chikkatur, Ji Zeng

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

Friday, September 11 2026, 00:00

Learn the Solid, Not the File: Canonical Inputs for Neural Networks on CAD Boundary Representations

Authors: Heinrich Jiang, Hager Yasser Mohamed, Alexander Hitt, Valeriia Lomakina, Henning Jiang, Jennifer Jang

Boundary representation (B-rep) is the standard format used by modern CAD systems for parametric 3D models. It turns out, the exact same solid can be represented by different B-reps: for example, two engineers using different operations, a geometry kernel rebuilding the file, and an export setting repartitioning faces will lead to different B-reps even though the underlying solid remains the same. We show that existing B-rep encoders are not robust to variation in the B-rep with the same solid on perturbations applied to standard benchmarks, naturally occurring variations inherent to CAD software, and differences in how designers model the same part via a human dataset we created in FreeCAD. The performance of popular B-rep encoders often collapses catastrophically. We propose the canonical region graph, an input representation whose nodes, features and coordinate frame are derived from the solid itself and show theoretical invariance guarantees on repartitioning and rigid motions. It matches the strongest baseline on standard benchmarks, and is stable under every perturbation we test.

Authors: Heinrich Jiang, Hager Yasser Mohamed, Alexander Hitt, Valeriia Lomakina, Henning Jiang, Jennifer Jang

Boundary representation (B-rep) is the standard format used by modern CAD systems for parametric 3D models. It turns out, the exact same solid can be represented by different B-reps: for example, two engineers using different operations, a geometry kernel rebuilding the file, and an export setting repartitioning faces will lead to different B-reps even though the underlying solid remains the same. We show that existing B-rep encoders are not robust to variation in the B-rep with the same solid on perturbations applied to standard benchmarks, naturally occurring variations inherent to CAD software, and differences in how designers model the same part via a human dataset we created in FreeCAD. The performance of popular B-rep encoders often collapses catastrophically. We propose the canonical region graph, an input representation whose nodes, features and coordinate frame are derived from the solid itself and show theoretical invariance guarantees on repartitioning and rigid motions. It matches the strongest baseline on standard benchmarks, and is stable under every perturbation we test.

Friday, September 11 2026, 00:00

Almost Linear Universal Point Sets for Planar Graphs

Authors: Taylor Gordon

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

Authors: Taylor Gordon

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

Friday, September 11 2026, 00:00

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

Authors: Édouard Bonnet, Yeonsu Chang

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milanič--Štorgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

Authors: Édouard Bonnet, Yeonsu Chang

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milanič--Štorgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

Friday, September 11 2026, 00:00

Tight Time-Space Lower Bounds for Collision Finding and Element Distinctness under Label Symmetry

Authors: Frédéric Magniez, Sebastian Zur

How much memory is needed to retain the quantum speedup for collision finding? For a uniformly random function $f:[N]\to [N]$, the BHT algorithm finds a collision using $O(N^{1/3})$ queries and a quantumly accessible classical table containing $O(N^{1/3})$ input-output pairs, whereas a logarithmic-space Grover search uses $O(\sqrt N)$ queries. Determining the optimal query-space tradeoff between these extremes remains a major open problem. We resolve this equation within the class of label-symmetric algorithms, which treat the function $f$'s output labels as interchangeable. We prove that such algorithm that makes $T$ queries, uses $S$ qubits, and finds a collision in a uniformly random function $f:[M]\to [N]$ with constant probability satisfies $$T=Ω(N^{1/3}) \qquad\text{and}\qquad T^2S=Ω(N\log N).$$ For the setting where $M=N$, these bounds are matched by a space-efficient implementation of the BHT algorithm. As a consequence of our tradeoff, any label-symmetric algorithm for the search version of Element Distinctness on $f: [n] \to [n^2]$ must satisfy $$T=Ω(n^{2/3}) \qquad\text{and}\qquad T^2S=Ω(n^2\log n),$$ matching Ambainis's quantum walk. Thus, both tradeoffs are optimal within the class of label-symmetric algorithms. To prove these results, we develop a space-sensitive version of the compressed oracle technique. The compressed oracle records the information learned by the algorithm in an evolving superposition of databases. Using label symmetry and representation theory, we show that an algorithm using $S$ qubits can effectively retain information about only $O(S/\log N)$ collision-free database entries. Substituting this estimate into the compressed oracle technique yields the stated tradeoffs.

Authors: Frédéric Magniez, Sebastian Zur

How much memory is needed to retain the quantum speedup for collision finding? For a uniformly random function $f:[N]\to [N]$, the BHT algorithm finds a collision using $O(N^{1/3})$ queries and a quantumly accessible classical table containing $O(N^{1/3})$ input-output pairs, whereas a logarithmic-space Grover search uses $O(\sqrt N)$ queries. Determining the optimal query-space tradeoff between these extremes remains a major open problem. We resolve this equation within the class of label-symmetric algorithms, which treat the function $f$'s output labels as interchangeable. We prove that such algorithm that makes $T$ queries, uses $S$ qubits, and finds a collision in a uniformly random function $f:[M]\to [N]$ with constant probability satisfies $$T=Ω(N^{1/3}) \qquad\text{and}\qquad T^2S=Ω(N\log N).$$ For the setting where $M=N$, these bounds are matched by a space-efficient implementation of the BHT algorithm. As a consequence of our tradeoff, any label-symmetric algorithm for the search version of Element Distinctness on $f: [n] \to [n^2]$ must satisfy $$T=Ω(n^{2/3}) \qquad\text{and}\qquad T^2S=Ω(n^2\log n),$$ matching Ambainis's quantum walk. Thus, both tradeoffs are optimal within the class of label-symmetric algorithms. To prove these results, we develop a space-sensitive version of the compressed oracle technique. The compressed oracle records the information learned by the algorithm in an evolving superposition of databases. Using label symmetry and representation theory, we show that an algorithm using $S$ qubits can effectively retain information about only $O(S/\log N)$ collision-free database entries. Substituting this estimate into the compressed oracle technique yields the stated tradeoffs.

Friday, September 11 2026, 00:00

A Reusable Framework for Robust Approximation Algorithms in the Interval Uncertainty Model

Authors: Klasing Ralf, Mömke Tobias, Naquin Émile

Robust optimization under interval uncertainty aims to compute solutions that perform well on a range of scenarios that are described by interval-constrained costs. In this paper, we revisit a framework introduced by Ganesh, Maggs and Panigrahi in 2020 to study the robust optimization of NP-hard problems under interval uncertainty. We start by generalizing a result in the $\ell=0$ case, which transforms a category of approximation algorithms into a robust approximation algorithm. Furthermore, in the general case, we provide a theorem that turns any local search-based approximation algorithm into a robust approximation algorithm under three newly formalized conditions over the moves of the local search algorithm. We then use this result to present the first robust approximation algorithm for Weighted $k$-Set Cover, the third NP-hard problem known to admit a robust approximation, and the first since the publication of Ganesh, Maggs and Panigrahi's paper.

Authors: Klasing Ralf, Mömke Tobias, Naquin Émile

Robust optimization under interval uncertainty aims to compute solutions that perform well on a range of scenarios that are described by interval-constrained costs. In this paper, we revisit a framework introduced by Ganesh, Maggs and Panigrahi in 2020 to study the robust optimization of NP-hard problems under interval uncertainty. We start by generalizing a result in the $\ell=0$ case, which transforms a category of approximation algorithms into a robust approximation algorithm. Furthermore, in the general case, we provide a theorem that turns any local search-based approximation algorithm into a robust approximation algorithm under three newly formalized conditions over the moves of the local search algorithm. We then use this result to present the first robust approximation algorithm for Weighted $k$-Set Cover, the third NP-hard problem known to admit a robust approximation, and the first since the publication of Ganesh, Maggs and Panigrahi's paper.

Friday, September 11 2026, 00:00

Convex Optimization with Nested Evolving Feasible Sets (CONES) under Time-Varying Loss Functions

Authors: Rahul Vaze

Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function $f$ remains fixed but the feasible region evolves over time as a nested sequence $S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T$. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \emph{nested convex body chasing} (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost, respectively, for any $β\in [0,1)$, over a time horizon of $T$. We also show that any {\it weakly adaptive} online algorithm with $O(T^β)$ regret has a movement cost of $Ω\left(T^{\frac{1-β}{2}}\right)$ for any $β\in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear {\it anytime} regret has a movement cost of $Ω\left(\log T\right)$.

Authors: Rahul Vaze

Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function $f$ remains fixed but the feasible region evolves over time as a nested sequence $S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T$. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \emph{nested convex body chasing} (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost, respectively, for any $β\in [0,1)$, over a time horizon of $T$. We also show that any {\it weakly adaptive} online algorithm with $O(T^β)$ regret has a movement cost of $Ω\left(T^{\frac{1-β}{2}}\right)$ for any $β\in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear {\it anytime} regret has a movement cost of $Ω\left(\log T\right)$.

Friday, September 11 2026, 00:00

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Authors: Tomohiro Koana, Soh Kumabe

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

Authors: Tomohiro Koana, Soh Kumabe

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

Friday, September 11 2026, 00:00

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

Authors: Dingding Dong, Vishesh Jain

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

Authors: Dingding Dong, Vishesh Jain

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

Friday, September 11 2026, 00:00

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Authors: Ekene Ezeunala, Agastya Vibhuti Jha, Haotian Jiang

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(σ_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $σ_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

Authors: Ekene Ezeunala, Agastya Vibhuti Jha, Haotian Jiang

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(σ_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $σ_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

Friday, September 11 2026, 00:00

Navigating Small-World Networks with Distance Predictions

Authors: Ladan Kian, Ming Ming Tan, Dariusz Kowalski

The small-world phenomenon was given an algorithmic foundation by Kleinberg, who showed that in an augmented $k$-dimensional lattice a decentralized greedy algorithm delivers a message in $O(\log^2 n)$ expected steps. We study predicted-greedy routing, in which a mobile agent forwarding the message moves at each step to the neighbor minimizing a noisy $(\varepsilon,δ)$-prediction of its distance to the target, redrawn at every step from an oracle conditioned on the full routing history. Two cases arise from what this agent can observe. An agent with the coordinate awareness can still compute lattice distance exactly, but not graph distance in the shortcut-augmented network, since that depends on the shortcuts of nodes it has not yet visited; given an $(\varepsilon,δ)$-prediction of graph distance, information the classical model never supplies, it achieves expected delivery time $O(\log n/(1-4k\varepsilonδ))$, an asymptotic improvement over $Θ(\log^2 n)$. An agent with no coordinate awareness at all, the natural model for a privacy-preserving network whose nodes never disclose their coordinates, cannot compute even lattice distance; given an $(\varepsilon,δ)$-prediction of lattice distance instead, it still reaches the target in $O(n/(1-4k\varepsilonδ))$ expected steps. Together these results show that a modest amount of predicted information, of the right kind, is enough to accelerate decentralized routing well below Kleinberg's classical bound, and that even when nodes reveal no coordinates at all, reliable delivery remains achievable.

Authors: Ladan Kian, Ming Ming Tan, Dariusz Kowalski

The small-world phenomenon was given an algorithmic foundation by Kleinberg, who showed that in an augmented $k$-dimensional lattice a decentralized greedy algorithm delivers a message in $O(\log^2 n)$ expected steps. We study predicted-greedy routing, in which a mobile agent forwarding the message moves at each step to the neighbor minimizing a noisy $(\varepsilon,δ)$-prediction of its distance to the target, redrawn at every step from an oracle conditioned on the full routing history. Two cases arise from what this agent can observe. An agent with the coordinate awareness can still compute lattice distance exactly, but not graph distance in the shortcut-augmented network, since that depends on the shortcuts of nodes it has not yet visited; given an $(\varepsilon,δ)$-prediction of graph distance, information the classical model never supplies, it achieves expected delivery time $O(\log n/(1-4k\varepsilonδ))$, an asymptotic improvement over $Θ(\log^2 n)$. An agent with no coordinate awareness at all, the natural model for a privacy-preserving network whose nodes never disclose their coordinates, cannot compute even lattice distance; given an $(\varepsilon,δ)$-prediction of lattice distance instead, it still reaches the target in $O(n/(1-4k\varepsilonδ))$ expected steps. Together these results show that a modest amount of predicted information, of the right kind, is enough to accelerate decentralized routing well below Kleinberg's classical bound, and that even when nodes reveal no coordinates at all, reliable delivery remains achievable.

Friday, September 11 2026, 00:00

Lower Bounds for Private Graph Optimization Problems using Reconstruction Attacks

Authors: Jacob Imola, Rasmus Pagh, Lukas Retschmeier

This paper studies fundamental graph optimization problems under differential privacy (DP) and shows new, reconstruction-based lower bounds. We consider a graph $G = (V, E, \vec{w})$ where the vertex set $V$ and edges $E$ are public and the weights $\mathbf{w}:E\rightarrow \mathbb{R}$ must be kept differentially private under an $\ell_1$ neighboring relation. For the problems of releasing a minimum-weight spanning tree and a minimum-weight perfect matching, we show new, tight error bounds of $Ω(n\cdot\log(m/n)/ε)$ on worst-case graphs with $n$ vertices and $m>2n$ edges. The upper bounds are known pure DP algorithms while the new lower bound holds even under approximate $(\varepsilon,δ)$-DP as long as $δ\leq (n/m)^{Ω(1)}$. Our lower bounds improve the $Ω(n/ε)$ lower bounds of Sealfon (PODS~'16). The fact that approximate DP does not reduce error for MST under the $\ell_1$ neighboring relation contrasts with the recent upper bound of Pagh et al. (PODS~'25) which shows that approximate DP allows much better error under the $\ell_\infty$ neighboring relation. Going beyond worst-case graphs, we give lower bounds for large families of sparse graphs with expansion properties. We show a lower bound of $Ω(n / ε)$ for the minimum spanning tree for any graph where the minimum cut is at least $Ω(\log(n))$. Finally, we consider the problem of private hierarchical clustering under Dasgupta's cost function (STOC~'16) and show the first approximate DP lower bound parameterized by the minimum weight of a balanced cut. This extends lower bounds of Deng et al. (ICLR~'25) to general graphs and to approximate DP.

Authors: Jacob Imola, Rasmus Pagh, Lukas Retschmeier

This paper studies fundamental graph optimization problems under differential privacy (DP) and shows new, reconstruction-based lower bounds. We consider a graph $G = (V, E, \vec{w})$ where the vertex set $V$ and edges $E$ are public and the weights $\mathbf{w}:E\rightarrow \mathbb{R}$ must be kept differentially private under an $\ell_1$ neighboring relation. For the problems of releasing a minimum-weight spanning tree and a minimum-weight perfect matching, we show new, tight error bounds of $Ω(n\cdot\log(m/n)/ε)$ on worst-case graphs with $n$ vertices and $m>2n$ edges. The upper bounds are known pure DP algorithms while the new lower bound holds even under approximate $(\varepsilon,δ)$-DP as long as $δ\leq (n/m)^{Ω(1)}$. Our lower bounds improve the $Ω(n/ε)$ lower bounds of Sealfon (PODS~'16). The fact that approximate DP does not reduce error for MST under the $\ell_1$ neighboring relation contrasts with the recent upper bound of Pagh et al. (PODS~'25) which shows that approximate DP allows much better error under the $\ell_\infty$ neighboring relation. Going beyond worst-case graphs, we give lower bounds for large families of sparse graphs with expansion properties. We show a lower bound of $Ω(n / ε)$ for the minimum spanning tree for any graph where the minimum cut is at least $Ω(\log(n))$. Finally, we consider the problem of private hierarchical clustering under Dasgupta's cost function (STOC~'16) and show the first approximate DP lower bound parameterized by the minimum weight of a balanced cut. This extends lower bounds of Deng et al. (ICLR~'25) to general graphs and to approximate DP.

Friday, September 11 2026, 00:00

Free-Probabilistic State Evolution and Random Matrix Discrepancy

Authors: August Y. Chen, Ahmed El Alaoui

Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\rightarrow \infty, 2n/d^2 \rightarrow α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \in \{-1,+1\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for an explicit expression of the standard deviation $σ(α)<1$ for all $0<α<α_* \simeq 5.74$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025) in this interval.

Authors: August Y. Chen, Ahmed El Alaoui

Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\rightarrow \infty, 2n/d^2 \rightarrow α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \in \{-1,+1\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for an explicit expression of the standard deviation $σ(α)<1$ for all $0<α<α_* \simeq 5.74$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025) in this interval.

Friday, September 11 2026, 00:00

Matters of Degree

from Ben Recht

Every forecasting course must engage with the dappled world of probability.

Hi there, argmin readers! As the fall semester picks up, posting volume will, too. So I’m going to commit to writing short descriptive headers to help you sort through the different threads. Today’s post is a live blog of Class 5 of my graduate seminar “Forecasting: A Critical Retrospective.” A table of contents is here.

The fun thing about teaching a new class is realizing in week three how your sequencing is already off. I do this every year, so I’m no longer surprised. At this point in my career, I’d put the probability that I’ll be disappointed with my syllabus before the official add-course deadline at 95%.

How did I get to that number?

So yes, though it’s not directly about forecasting, the first mathematical lecture of this class should have been about probability. Because probability is unavoidable in forecasting, and it is used far too casually for my tastes. We saw this in the weather example where outputs of data assimilation were interpreted as “probability distributions” over the state of the atmosphere. Samples from this distribution were used to create a sample of future possibilities. Frequencies of these future possibilities were turned into beliefs about whether it will rain on Sunday.

All of these probabilities are sort of different, right? Some are about counts, some are about beliefs, and somehow we move between the two as if there is a well-specified set of formal rules for doing so.

I think every class on applied probability, including the undergrad ones, should call out this slippery transmutation between frequency and belief, and I’m particularly fond of the development in Paul Meehl’s course on Philosophical Psychology.1 Meehl follows Rudolf Carnap, who explicitly distinguishes the two kinds of probability.

Probability 1, sometimes called logical probability, relates propositions with beliefs. It quantifies how much credence we should give a hypothesis based on the observations and facts laid before us. Probability 1 represents the certainty of a logical proposition being true, the doubt that something in the past happened, the likelihood that future events will occur, or a person’s internal beliefs about the world.

Probability 2 concerns frequencies of events. It more or less just amounts to counting, measuring the relative frequency of some property in a set of objects. Probability 2 might quantify the relative frequencies of occurrences that currently exist in the world, but it can also describe the relative frequencies of hypothetical infinite populations.

Probability 2 is usually the one we start with and teach and never problematize, and then we casually jump to using Probability 1 in our thinking, writing, and experimenting without realizing it. This is because they both obey the same axioms of probability.

Let’s say I have a bunch of marbles in an urn. I want to describe the frequencies with which certain properties of those marbles hold. The following things are true for any subset of marbles:

Any subset has proportion greater than or equal to 0
If I take the entire set, the proportion equals 1.
If I take two nonoverlapping subsets, the proportion of their union is equal to the sum of their proportions.

These are Kolmogorov’s axioms of probability. Item 1 is nonnegativity, Item 2 is unit measure, Item 3 is additivity.

Probability 2 obviously obeys Kolmogorov’s axioms, but what about Probability 1? Well, we can sort of force it to have those properties too.

Any syntactically valid statement has probability greater than or equal to zero.
Any statement that is certain or tautological has probability 1.
If two statements describe mutually exclusive outcomes, then the probability of either one or both of the outcomes is equal to the sum of the individual probabilities.

Lo and behold, statements of belief seem to also obey Kolmogorov’s Axioms. These assertions might feel a bit less obvious and certain than marble counting. You can’t get a tangible grip on why beliefs should obey Kolmogorov’s axioms because beliefs live in your head. You can check the properties of Probability 2 on a set of marbles. You can never check whether the axioms work for Probability 1.

But there are strategic reasons for using probabilities when quantifying beliefs. In last Thursday’s lecture, we saw that if you are scored using the Brier Score, then when your forecasts don’t obey the axioms of probability, there is always another set of forecasts that achieves a higher place on the leaderboard. If instead of being a forecaster, you’re a degenerate gambler, there are arguments about dealing with bookies that motivate the same probabilistic rules. As Dennis Lindley put it, probability is inevitable once you try to quantitatively evaluate belief.

I think we should also distinguish a third kind of probability, Probability 0, to describe the formal language of mathematical probability. This is probability whose referent is neither frequencies nor beliefs but mathematics itself. In this case, we’d consider the following to all be Probability 0:

Any finite list of numbers that is nonnegative and sums to one
Any infinite list of numbers that is nonnegative and whose infinite sum converges to one
Any nonnegative function on the unit interval whose integral is equal to one

These sorts of mathematical structures arise a lot when we’re doing calculations. And whenever people find a convenient mathematical application of Probability 0, they tend to find a convenient application of that structure in Probability 1 or Probability 2. We see Probability 0 everywhere. It only becomes metaphysical when we attach some meaning to it off the chalkboard.

When we build computational forecasts, we have to play with all three kinds of probability. We saw this last time: We need Probability 2 because the best predictions correspond to the rates of outcomes in similar future events. We need Probability 1, or else our forecasts are incoherent. We need Probability 0 to write and reason about algorithms that analyze noisy data. But how we tie the three together is not given. There is no god-given algorithm of prognostication that we can derive from Kolmogorov’s axioms, and techniques and conventions vary between disciplines. Which scoring rule are we using? Are we insisting on building calibrated forecasts? What do our forecasts do? These questions determine how we work with probability.

Subscribe now

1

The videos are here, and my blog series engaging with the course is here.

By Ben Recht

Thursday, September 10 2026, 14:32

TR26-173 | Matrix identities are hard: Fast blackbox PIT for noncommutative exponential-size constant-depth homogeneous circuits | Foram Lakhani, Nitin Saxena

from ECCC Papers

In this work we give a randomized blackbox polynomial identity testing (PIT) algorithm for constant-depth homogeneous noncommutative circuits, with poly-logarithmic time complexity in the circuit size. In fact, we show that the polynomial computable by such a depth-$(\Delta-1)$ circuit of size $s$ cannot be a polynomial identity for the $O(\log ^{\Delta}s)$-dimension matrix algebra $\mathcal{M}_{O(\log ^{\Delta}s)}(\mathbb{F})$, for a sufficiently large field $\mathbb{F}$. This represents progress toward answering the question raised by Arvind, Joglekar, Mukhopadhyay, and Raja (2019), and Bogdanov and Wee (2005) to find efficient randomized blackbox PIT algorithm for circuits that output polynomials with $doubly$ exponential sparsity. Arvind, Joglekar, Mukhopadhyay, and Raja (2019) and Bharadwaj and S. Raja (2025) defined constant-depth $+$-regular circuit and gave a PIT for it. Their model of size $S$ is simulated by our model of size $\exp(S)$; hence, we recover their result. Because of the regularity condition their model is highly restrictive than ours. We decompose a depth-$\Delta$ formula into its constituent depth-$(\Delta - 1)$ subformulas and analyze them concurrently using Hadamard algebra, subsequently translating these findings back to the depth-$\Delta$ setting. We reduce the multivariate case to the bivariate case involving the noncommuting variables $x$ and $y$. Finally, we apply a transformation to $x$ and $y$ to ensure that the nonzero witness coefficient is associated with a monomial of exponentially-lower $y$-degree. We call this the phenomenon of $noncommutative\ low-degree\ concentration$.

Thursday, September 10 2026, 14:01

On the Limits of Quantum Multiparty Simultaneous Communication

Authors: Pedro Montealegre, Ivan Rapaport, Jorge Valenzuela

The Simultaneous Message Passing (SMP) model provides a fundamental framework for comparing classical and quantum communication. For two players, Gavinsky et al. (STOC 2006) established a separation underlying the incomparability of shared randomness and quantum communication: \textsc{Index Coordination} needs $O(\log n)$ public-coin bits but $Ω(n^{1/3})$ bounded-error qubits. In this work, we establish a multiparty exponential separation through $\operatorname{IC}_{k,n}$, a natural $k$-party generalization of \textsc{Index Coordination}. Public-coin protocols solve it unambiguously with maximum message length $O(\log n)$ bits. In contrast, quantum SMP protocols without shared entanglement or public coins require maximum message length $Ω(n^{1-1/k})$ qubits in the unambiguous regime and $Ω(n^{(k-1)/(k+1)})$ qubits in the bounded-error regime. A classical private-coin protocol matches the unambiguous bound, so quantum communication provides no asymptotic advantage over private randomness in this regime. For fixed error parameters, all constants are independent of $k$, establishing the exponential separation for every integer-valued function $k=k(n)\ge2$, without restricting its growth. Both quantum lower bounds become $Ω(n)$ when $k\ge c\log n$ for any fixed $c>0$, matching the full-input protocol and yielding tight linear complexity in both regimes. Our results demonstrate that quantum superposition cannot efficiently simulate the coordination afforded by public randomness, extending this separation to arbitrary $k$. To bound success probabilities for multiparty product states, we prove an exact factorization theorem for unambiguous quantum state identification, which may be of independent mathematical interest.

Authors: Pedro Montealegre, Ivan Rapaport, Jorge Valenzuela

The Simultaneous Message Passing (SMP) model provides a fundamental framework for comparing classical and quantum communication. For two players, Gavinsky et al. (STOC 2006) established a separation underlying the incomparability of shared randomness and quantum communication: \textsc{Index Coordination} needs $O(\log n)$ public-coin bits but $Ω(n^{1/3})$ bounded-error qubits. In this work, we establish a multiparty exponential separation through $\operatorname{IC}_{k,n}$, a natural $k$-party generalization of \textsc{Index Coordination}. Public-coin protocols solve it unambiguously with maximum message length $O(\log n)$ bits. In contrast, quantum SMP protocols without shared entanglement or public coins require maximum message length $Ω(n^{1-1/k})$ qubits in the unambiguous regime and $Ω(n^{(k-1)/(k+1)})$ qubits in the bounded-error regime. A classical private-coin protocol matches the unambiguous bound, so quantum communication provides no asymptotic advantage over private randomness in this regime. For fixed error parameters, all constants are independent of $k$, establishing the exponential separation for every integer-valued function $k=k(n)\ge2$, without restricting its growth. Both quantum lower bounds become $Ω(n)$ when $k\ge c\log n$ for any fixed $c>0$, matching the full-input protocol and yielding tight linear complexity in both regimes. Our results demonstrate that quantum superposition cannot efficiently simulate the coordination afforded by public randomness, extending this separation to arbitrary $k$. To bound success probabilities for multiparty product states, we prove an exact factorization theorem for unambiguous quantum state identification, which may be of independent mathematical interest.

Thursday, September 10 2026, 00:00

On the Tightness of Standard Relaxations for Mixed-Integer Bilevel Linear Programs

Authors: Sergey S. Ketkov, Oleg A. Prokopyev

Exact algorithms for solving mixed-integer bilevel linear programs (MIBLPs) typically rely on sequences of lower and upper bounds that converge to the optimal value. These procedures are commonly initialized using the single-level relaxation (SLR), obtained by omitting the follower's optimality condition and solving the resulting single-level optimization problem. In this paper, we investigate whether, for broad classes of MIBLPs, the resulting standard bounds admit uniform improvements that can be computed within the same computational complexity regime. For pure continuous bilevel linear programs, we show that, unless $P = NP$, neither the SLR-based lower bound nor its associated upper bound can be uniformly improved in polynomial time, even for the class of min-max problems. We then extend this analysis to the class of pure integer min-max bilevel linear programs under the assumption that the polynomial hierarchy does not collapse. First, we show that the continuous relaxation of the SLR admits no uniform polynomial-time computable improvement. We then prove that neither the SLR itself nor its associated upper bound admits a uniform improvement by a polynomial-time algorithm with access to a mixed-integer linear programming (MILP) oracle. Importantly, this rules out uniform improvements by iterative MILP-based approaches, including cutting-plane-based and decomposition algorithms. Overall, our results demonstrate that the SLR-based bounds are, in a complexity-theoretic sense, unimprovable systematically within their natural computational regimes.

Authors: Sergey S. Ketkov, Oleg A. Prokopyev

Exact algorithms for solving mixed-integer bilevel linear programs (MIBLPs) typically rely on sequences of lower and upper bounds that converge to the optimal value. These procedures are commonly initialized using the single-level relaxation (SLR), obtained by omitting the follower's optimality condition and solving the resulting single-level optimization problem. In this paper, we investigate whether, for broad classes of MIBLPs, the resulting standard bounds admit uniform improvements that can be computed within the same computational complexity regime. For pure continuous bilevel linear programs, we show that, unless $P = NP$, neither the SLR-based lower bound nor its associated upper bound can be uniformly improved in polynomial time, even for the class of min-max problems. We then extend this analysis to the class of pure integer min-max bilevel linear programs under the assumption that the polynomial hierarchy does not collapse. First, we show that the continuous relaxation of the SLR admits no uniform polynomial-time computable improvement. We then prove that neither the SLR itself nor its associated upper bound admits a uniform improvement by a polynomial-time algorithm with access to a mixed-integer linear programming (MILP) oracle. Importantly, this rules out uniform improvements by iterative MILP-based approaches, including cutting-plane-based and decomposition algorithms. Overall, our results demonstrate that the SLR-based bounds are, in a complexity-theoretic sense, unimprovable systematically within their natural computational regimes.

Thursday, September 10 2026, 00:00

When Does a Quantum Speedup Survive End-to-End?

Authors: Pablo Herrero Gómez, Antonio Jimeno Morenilla, David Muñoz Hernández, Higinio Mora Mora

Primitive quantum speedups are interface-relative: they depend on the input access used to run the primitive and on the output contract used to consume its state or samples. This paper introduces a transcript-level admissibility relation $A_M\preceq_{\mathrm{int}}A_Q$, defined relative to the declared implementation package of the quantum interface. It identifies which adaptive classical access transcripts that same package licenses, with all setup, transcript-generation, and precision overheads charged. The main application is an operational audit for normalized-Betti estimation in clique-complex TDA, separating three declared-interface regimes. Reversible indexed simplex interfaces certify matched classical simplex sampling and local Laplacian row access by evaluating their reversible routines on single computational branches. Membership-based preparations induce a rejection route of overhead $\binom{n}{k+1}/|S_k|$. Abstract spectral or block-encoding interfaces require an accompanying implementation package, transcript reduction, or shared representation. Under the indexed certificate and interface closure, the end-to-end cost is fixed by the imported estimator's spectral dependence on the gap $γ$; the concretely realized bounded-treewidth family already admits exact $\mathrm{poly}(n)$ classical Betti computation by rank over $\mathbb{Q}$. A low-rank separation supports the role of access and output contracts.

Authors: Pablo Herrero Gómez, Antonio Jimeno Morenilla, David Muñoz Hernández, Higinio Mora Mora

Primitive quantum speedups are interface-relative: they depend on the input access used to run the primitive and on the output contract used to consume its state or samples. This paper introduces a transcript-level admissibility relation $A_M\preceq_{\mathrm{int}}A_Q$, defined relative to the declared implementation package of the quantum interface. It identifies which adaptive classical access transcripts that same package licenses, with all setup, transcript-generation, and precision overheads charged. The main application is an operational audit for normalized-Betti estimation in clique-complex TDA, separating three declared-interface regimes. Reversible indexed simplex interfaces certify matched classical simplex sampling and local Laplacian row access by evaluating their reversible routines on single computational branches. Membership-based preparations induce a rejection route of overhead $\binom{n}{k+1}/|S_k|$. Abstract spectral or block-encoding interfaces require an accompanying implementation package, transcript reduction, or shared representation. Under the indexed certificate and interface closure, the end-to-end cost is fixed by the imported estimator's spectral dependence on the gap $γ$; the concretely realized bounded-treewidth family already admits exact $\mathrm{poly}(n)$ classical Betti computation by rank over $\mathbb{Q}$. A low-rank separation supports the role of access and output contracts.

Thursday, September 10 2026, 00:00

Small-Bias Quantum Approximate Counting via the Multiplicative Adversary Method

Authors: Albert Lin, Han-Hsuan Lin

We study the two-weight decision version of quantum approximate counting: given oracle access to $x\in\{0,1\}^N$, distinguish $|x|=M$ from $|x|=M+Δ$ with success probability $1/2+ζ$. Using the multiplicative adversary method, we prove $Ω\left(\max\left\{ζ\sqrt{(N-M)(M+Δ)}/Δ,\sqrt{ζN/Δ}\right\}\right)$. The same parameter dependence follows from the polynomial-method characterization of the two-layer symmetric function by Podder, Yao, and Ye. Our contribution is a multiplicative-adversary derivation that tracks the progress produced by individual oracle queries. For the first term, after complementing the input if necessary, we assume $M+Δ\le N-M$. We use the Hamming-layer subspaces from the eigenspace method of Ambainis, Spalek, and de Wolf and compose their adjacent-layer unitary maps to relate the two nonadjacent promise layers. After fixing the queried coordinate, the analysis block-diagonalizes into four-dimensional subspaces. An exact calculation of the one-query progress ratio gives the first lower bound. The same estimate also implies $\left\|(I-\widehatΠ_{\mathrm{bad}})\lvertΨ^T\rangle\right\|^2=O\left(T^2Δ^2/((N-M)(M+Δ))\right)$ for the coherent input superposition used in the adversary argument. For the second term, we prove directly using a three-eigenvalue multiplicative adversary that unique OR on $n$ bits with success probability $1/2+ζ$ requires $Ω(\sqrt{ζn})$ queries, and then reduce unique OR to the two-weight counting problem.

Authors: Albert Lin, Han-Hsuan Lin

We study the two-weight decision version of quantum approximate counting: given oracle access to $x\in\{0,1\}^N$, distinguish $|x|=M$ from $|x|=M+Δ$ with success probability $1/2+ζ$. Using the multiplicative adversary method, we prove $Ω\left(\max\left\{ζ\sqrt{(N-M)(M+Δ)}/Δ,\sqrt{ζN/Δ}\right\}\right)$. The same parameter dependence follows from the polynomial-method characterization of the two-layer symmetric function by Podder, Yao, and Ye. Our contribution is a multiplicative-adversary derivation that tracks the progress produced by individual oracle queries. For the first term, after complementing the input if necessary, we assume $M+Δ\le N-M$. We use the Hamming-layer subspaces from the eigenspace method of Ambainis, Spalek, and de Wolf and compose their adjacent-layer unitary maps to relate the two nonadjacent promise layers. After fixing the queried coordinate, the analysis block-diagonalizes into four-dimensional subspaces. An exact calculation of the one-query progress ratio gives the first lower bound. The same estimate also implies $\left\|(I-\widehatΠ_{\mathrm{bad}})\lvertΨ^T\rangle\right\|^2=O\left(T^2Δ^2/((N-M)(M+Δ))\right)$ for the coherent input superposition used in the adversary argument. For the second term, we prove directly using a three-eigenvalue multiplicative adversary that unique OR on $n$ bits with success probability $1/2+ζ$ requires $Ω(\sqrt{ζn})$ queries, and then reduce unique OR to the two-weight counting problem.

Thursday, September 10 2026, 00:00

NP-Hardness of the $H$-Free Edge-Deletion Problem

Authors: Lior Gishboliner, Ethan Honest

For a graph $H$, the $H$-freeness edge-deletion problem is the algorithmic problem of finding, for an input graph $G$, the minimum number of edges of $G$ whose deletion turns $G$ into an $H$-free graph. We show that for every graph $H$ containing a cycle, this problem is NP-hard. This proves a conjecture of Gishboliner, Levanzov and Shapira, and completes the characterization of the complexity of the $H$-freeness edge-deletion problem, answering a question of Alon, Shapira and Sudakov.

Authors: Lior Gishboliner, Ethan Honest

For a graph $H$, the $H$-freeness edge-deletion problem is the algorithmic problem of finding, for an input graph $G$, the minimum number of edges of $G$ whose deletion turns $G$ into an $H$-free graph. We show that for every graph $H$ containing a cycle, this problem is NP-hard. This proves a conjecture of Gishboliner, Levanzov and Shapira, and completes the characterization of the complexity of the $H$-freeness edge-deletion problem, answering a question of Alon, Shapira and Sudakov.

Thursday, September 10 2026, 00:00

A Note on the Point-Clothoid Distance Algorithm

Authors: Haibin Ye, Hao Ge, Gong Cheng

Computing the closest point on a clothoid is a recurring task in geometric design, road and railway alignment, and path planning. The efficient algorithm of Frego and Bertolazzi addresses this problem, but its candidate-selection analysis assumes at most one local minimum per search interval. We exhibit admissible configurations with two local minima, raising the question of whether the existing strategy accounts for every possible minimum. Using the geometry of the clothoid evolute, we prove that, for any query point and any proper no-inflection planar clothoid segment with tangent-angle variation at most $2π$, the squared-distance function has at most three stationary points; if all three are local extrema, their order is min-max-min. This establishes the completeness of the original candidate-selection logic beyond the one-minimum premise. It also shows that no interior search is needed when neither endpoint derivative test is active, allowing unnecessary midpoint searches to be omitted while retaining numerical fallback. Numerical experiments demonstrate reductions in iteration count and evaluation time.

Authors: Haibin Ye, Hao Ge, Gong Cheng

Computing the closest point on a clothoid is a recurring task in geometric design, road and railway alignment, and path planning. The efficient algorithm of Frego and Bertolazzi addresses this problem, but its candidate-selection analysis assumes at most one local minimum per search interval. We exhibit admissible configurations with two local minima, raising the question of whether the existing strategy accounts for every possible minimum. Using the geometry of the clothoid evolute, we prove that, for any query point and any proper no-inflection planar clothoid segment with tangent-angle variation at most $2π$, the squared-distance function has at most three stationary points; if all three are local extrema, their order is min-max-min. This establishes the completeness of the original candidate-selection logic beyond the one-minimum premise. It also shows that no interior search is needed when neither endpoint derivative test is active, allowing unnecessary midpoint searches to be omitted while retaining numerical fallback. Numerical experiments demonstrate reductions in iteration count and evaluation time.

Thursday, September 10 2026, 00:00

A 2.37332-Competitive Algorithm for Online Square Packing with Gravity

Authors: Nichlas Langhoff Rasmussen

We consider online packing of axis-parallel squares into a unit-width strip under the Tetris and gravity constraints: An incoming square must be lowered from above along a monotonic downwards path until it reaches support from below. Fekete, Kamphans, and Schweer [Algorithmica, 2014] gave an algorithm with asymptotic competitive ratio $34/13\approx2.6154$ in this model. We present $\mathrm{AsymmetricSlots}$, a recursive algorithm based on splitting each slot into a wide and narrow subslot. The proof uses a local charging argument: squares that are large relative to its associated slot pay for the height they create with their own area, while smaller squares are balanced between the two subslots and may use a bounded temporary credit. For a suitable split parameter $p^\star$, we prove $\mathrm{AsymmetricSlots}{p^\star}(σ)\le 2.37332 \operatorname{OPT} (σ)+O(1)$ for every input sequence $σ$. Additionally, we show that the same framework gives an algorithm with asymptotic competitive ratio $O(κ)$ for rectangles of aspect ratio at most $κ$, and a matching $ Ω(κ)$ lower bound shows that the dependence on $κ$ is asymptotically optimal. For the square algorithm, we give a lower bound of $2$ on its asymptotic competitive ratio.

Authors: Nichlas Langhoff Rasmussen

We consider online packing of axis-parallel squares into a unit-width strip under the Tetris and gravity constraints: An incoming square must be lowered from above along a monotonic downwards path until it reaches support from below. Fekete, Kamphans, and Schweer [Algorithmica, 2014] gave an algorithm with asymptotic competitive ratio $34/13\approx2.6154$ in this model. We present $\mathrm{AsymmetricSlots}$, a recursive algorithm based on splitting each slot into a wide and narrow subslot. The proof uses a local charging argument: squares that are large relative to its associated slot pay for the height they create with their own area, while smaller squares are balanced between the two subslots and may use a bounded temporary credit. For a suitable split parameter $p^\star$, we prove $\mathrm{AsymmetricSlots}{p^\star}(σ)\le 2.37332 \operatorname{OPT} (σ)+O(1)$ for every input sequence $σ$. Additionally, we show that the same framework gives an algorithm with asymptotic competitive ratio $O(κ)$ for rectangles of aspect ratio at most $κ$, and a matching $ Ω(κ)$ lower bound shows that the dependence on $κ$ is asymptotically optimal. For the square algorithm, we give a lower bound of $2$ on its asymptotic competitive ratio.

Thursday, September 10 2026, 00:00

Overlap-Helly theorems

Authors: Andreas F. Holmsen, Alfredo Hubard

In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

Authors: Andreas F. Holmsen, Alfredo Hubard

In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

Thursday, September 10 2026, 00:00

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Authors: Vincent Despre, Auguste Gezalyan, Marc Pouget

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

Authors: Vincent Despre, Auguste Gezalyan, Marc Pouget

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

Thursday, September 10 2026, 00:00

iLogMap: Geodesic Polar Coordinates Parameterization with the Magnetic Laplacian

Authors: Tomás Banduc, Simone Pezzuto, Francisco Sahli Costabal

Geodesic polar coordinates (GPCs) provide an intrinsic parameterization over curved surfaces, but their accurate estimation remains challenging, particularly in the presence of anisotropic metrics, high curvature and complex topology. We introduce iLogMap, a method for computing GPCs in curved domains that recasts the angular component of the logarithmic map to a ground-state magnetic eigenproblem over the circumferential direction field of geodesic distance. Our method effortlessly extends to anisotropic metric tensors and solid volumes, enabling cylindrical and spherical parameterizations in tetrahedral meshes. Experiments on diverse shapes with varying genus confirm competitive angular accuracy and reduced metric distortion relative to heat-based methods, with improved performance on surfaces with boundary and domains with anisotropy. We demonstrate the utility of iLogMap in computational cardiology applications, where we use it to initialize spiral phases on atrial surfaces and estimate local activation patterns in ventricular models.

Authors: Tomás Banduc, Simone Pezzuto, Francisco Sahli Costabal

Geodesic polar coordinates (GPCs) provide an intrinsic parameterization over curved surfaces, but their accurate estimation remains challenging, particularly in the presence of anisotropic metrics, high curvature and complex topology. We introduce iLogMap, a method for computing GPCs in curved domains that recasts the angular component of the logarithmic map to a ground-state magnetic eigenproblem over the circumferential direction field of geodesic distance. Our method effortlessly extends to anisotropic metric tensors and solid volumes, enabling cylindrical and spherical parameterizations in tetrahedral meshes. Experiments on diverse shapes with varying genus confirm competitive angular accuracy and reduced metric distortion relative to heat-based methods, with improved performance on surfaces with boundary and domains with anisotropy. We demonstrate the utility of iLogMap in computational cardiology applications, where we use it to initialize spiral phases on atrial surfaces and estimate local activation patterns in ventricular models.

Thursday, September 10 2026, 00:00

On the Parameterized Complexity of Coloring Discovery

Authors: Eric Decker, Sebastian Siebertz

Coloring Discovery asks whether a possibly improper initial coloring can be made proper within a prescribed number of allowed changes. We study the parameterized complexity of three modification step models that were studied previously in the literature: recoloring one vertex (color flipping), swapping the colors of arbitrary vertices (color swapping), and swapping colors only across an edge (color sliding). For color flipping, we give exact fixed-parameter algorithms for the parameters vertex cover and distance to complete. For color swapping, we obtain fixed-parameter tractability for the parameter vertex cover plus the number of colors. Our lower bounds show W[1]-hardness for treedepth plus feedback vertex set in the color flipping model and for the number of colors plus bandwidth or distance to disjoint paths in the swapping and sliding models. All three variants remain NP-complete with four colors on graphs of diameter two.

Authors: Eric Decker, Sebastian Siebertz

Coloring Discovery asks whether a possibly improper initial coloring can be made proper within a prescribed number of allowed changes. We study the parameterized complexity of three modification step models that were studied previously in the literature: recoloring one vertex (color flipping), swapping the colors of arbitrary vertices (color swapping), and swapping colors only across an edge (color sliding). For color flipping, we give exact fixed-parameter algorithms for the parameters vertex cover and distance to complete. For color swapping, we obtain fixed-parameter tractability for the parameter vertex cover plus the number of colors. Our lower bounds show W[1]-hardness for treedepth plus feedback vertex set in the color flipping model and for the number of colors plus bandwidth or distance to disjoint paths in the swapping and sliding models. All three variants remain NP-complete with four colors on graphs of diameter two.

Thursday, September 10 2026, 00:00

Optimal Non-Adaptive Vantage Point Selection

Authors: Jie Gao, Nicole Wein, Chang Wu

We study the \emph{vantage point selection} problem, introduced by Ashvinkumar, Chowdhury, Gao, Goswami, Mitchell, and Polishchuk [WADS'25] to model the problem of estimating bottleneck capacities on the Internet. The input is a weighted undirected graph with unique shortest paths where every edge has a distinct unknown \emph{capacity}. When the algorithm \emph{queries} a vertex $v$, it reveals the minimum-capacity edge on the shortest path from $v$ to every other vertex reachable from $v$. The goal is to maximize the total number of revealed edges. The quality of an algorithm is measured by its competitive ratio against an optimal algorithm that knows all edge capacities a priori. We first consider the foundational single-query setting, where both the algorithm and the optimal algorithm are restricted to a single query. There is a trivial upper bound of $O(n)$ on the competitive ratio and the best known lower bound was $\tildeΩ(\sqrt{n})$. We provide an algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible competitive ratio is $\tildeΘ(n^{2/3})$. Furthermore, we extend our results to the general setting where the optimal algorithm is allowed $k$ queries and our algorithm is allowed $αk$ queries for $α\geq 1$. We present a randomized non-adaptive algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible expected competitive ratio for non-adaptive algorithms is the following surprisingly complex bound: $$ \tildeΘ\left( \min\left\{ \frac{n}{αk}, \max\left( \sqrt{\frac{n}α}, \frac{n^{2/3}}{αk^{1/3}} \right) \right\} \right). $$

Authors: Jie Gao, Nicole Wein, Chang Wu

We study the \emph{vantage point selection} problem, introduced by Ashvinkumar, Chowdhury, Gao, Goswami, Mitchell, and Polishchuk [WADS'25] to model the problem of estimating bottleneck capacities on the Internet. The input is a weighted undirected graph with unique shortest paths where every edge has a distinct unknown \emph{capacity}. When the algorithm \emph{queries} a vertex $v$, it reveals the minimum-capacity edge on the shortest path from $v$ to every other vertex reachable from $v$. The goal is to maximize the total number of revealed edges. The quality of an algorithm is measured by its competitive ratio against an optimal algorithm that knows all edge capacities a priori. We first consider the foundational single-query setting, where both the algorithm and the optimal algorithm are restricted to a single query. There is a trivial upper bound of $O(n)$ on the competitive ratio and the best known lower bound was $\tildeΩ(\sqrt{n})$. We provide an algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible competitive ratio is $\tildeΘ(n^{2/3})$. Furthermore, we extend our results to the general setting where the optimal algorithm is allowed $k$ queries and our algorithm is allowed $αk$ queries for $α\geq 1$. We present a randomized non-adaptive algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible expected competitive ratio for non-adaptive algorithms is the following surprisingly complex bound: $$ \tildeΘ\left( \min\left\{ \frac{n}{αk}, \max\left( \sqrt{\frac{n}α}, \frac{n^{2/3}}{αk^{1/3}} \right) \right\} \right). $$

Thursday, September 10 2026, 00:00

Introvert Clustering for Distributed Graph Algorithms

Authors: Yi-Jun Chang, Nima Dolatabadi

We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a $\left(\frac12-\varepsilon\right)$-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with $O(\log n)$ layers and weak diameter $O(\log n)$. We give two applications in the $\mathsf{LOCAL}$ model. For every constant $\varepsilon>0$, we obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for list $\left(\frac32+\varepsilon\right)Δ$-edge coloring on graphs of maximum degree $Δ\geqΔ_0(\varepsilon)$; for bipartite graphs, the result holds for all $Δ$. For every constant $0<\varepsilon<1/4$, we also obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for a $\left(\frac14-\varepsilon\right)$-locally balanced cut, where every vertex has at least a $\left(\frac14-\varepsilon\right)$-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in $O(\log^2 n)$ randomized rounds using Miller--Peng--Xu low-diameter clustering and a simple trimming procedure, and deterministically in $\widetilde O(\log^2 n)$ rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].

Authors: Yi-Jun Chang, Nima Dolatabadi

We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a $\left(\frac12-\varepsilon\right)$-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with $O(\log n)$ layers and weak diameter $O(\log n)$. We give two applications in the $\mathsf{LOCAL}$ model. For every constant $\varepsilon>0$, we obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for list $\left(\frac32+\varepsilon\right)Δ$-edge coloring on graphs of maximum degree $Δ\geqΔ_0(\varepsilon)$; for bipartite graphs, the result holds for all $Δ$. For every constant $0<\varepsilon<1/4$, we also obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for a $\left(\frac14-\varepsilon\right)$-locally balanced cut, where every vertex has at least a $\left(\frac14-\varepsilon\right)$-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in $O(\log^2 n)$ randomized rounds using Miller--Peng--Xu low-diameter clustering and a simple trimming procedure, and deterministically in $\widetilde O(\log^2 n)$ rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].

Thursday, September 10 2026, 00:00

Minimum-makespan completion and vertex selection leave the Wang-Sitters constant at 11/6

Authors: Adam Y. Shavit

The 11/6 worst-case constant of the Wang-Sitters rounding scheme, which a companion note establishes, can naturally be attributed to the freedom in Step 3, where an arbitrary valid slot matching is permitted. We show that eliminating that freedom does not improve the constant. A minimum-makespan completion oracle still has worst-case constant exactly 11/6 against the optimum; both natural 7/4 statements about it are false; and restricting Step 1 to vertices of the relaxation does not help. The loss therefore cannot be attributed solely to the freedom in Step 3. We also record what structure survives: a reduction confining every overload to two shapes, a seven-machine instance defeating the natural two-phase repair, and a strict 7/4 bound on the generalized three-path family.

Authors: Adam Y. Shavit

The 11/6 worst-case constant of the Wang-Sitters rounding scheme, which a companion note establishes, can naturally be attributed to the freedom in Step 3, where an arbitrary valid slot matching is permitted. We show that eliminating that freedom does not improve the constant. A minimum-makespan completion oracle still has worst-case constant exactly 11/6 against the optimum; both natural 7/4 statements about it are false; and restricting Step 1 to vertices of the relaxation does not help. The loss therefore cannot be attributed solely to the freedom in Step 3. We also record what structure survives: a reduction confining every overload to two shapes, a seven-machine instance defeating the natural two-phase repair, and a strict 7/4 bound on the generalized three-path family.

Thursday, September 10 2026, 00:00

A Sharp Barrier for Consistent Submodular Maximization: Any Improvement over $2-\sqrt{2}$ Entails Exponential Queries or Linear Recourse

Authors: Shi Fu, Qixin Zhang, Dacheng Tao

Consistent submodular maximization studies the tradeoff between solution quality and stability when elements arrive over time. For a monotone submodular objective, which models diminishing returns, an algorithm maintains a set of at most $k$ available elements and changes only $O(1)$ elements after each insertion. Dütting et al. [2025] established a tight $2/3$ approximation with unrestricted computation and a polynomial-time $0.51$ approximation. They left open at STOC 2025 whether efficient algorithms can match the offline $1-1/e$ guarantee. We resolve this problem by proving that the supremum approximation achievable with polynomially many value queries and worst-case constant recourse is \[ β=2-\sqrt2\approx0.5858<1-1/e. \] For every $\varepsilon>0$, our randomized algorithm attains $β-\varepsilon$ with $O(\varepsilon^{-2})$ changes per insertion. Any fixed improvement requires exponentially many queries before one critical insertion or linear recourse of $Ω(k)$ changes at that insertion, even with unlimited queries afterwards. This gap quantifies the cost of consistency: the current oracle hides which elements will be needed after an arrival. We also determine the exact curvature-dependent threshold $1-(\sqrt2-1)\vartheta$, attain $1-1/e-\varepsilon$ for weighted coverage with $O(\varepsilon^{-1})$ recourse, and separate the existence of universal future-price certificates from their efficient computation. Our algorithm has a bounded-bit polynomial-time implementation for polynomial-bit rational oracle answers; the lower bound uses only logarithmic-bit rational answers.

Authors: Shi Fu, Qixin Zhang, Dacheng Tao

Consistent submodular maximization studies the tradeoff between solution quality and stability when elements arrive over time. For a monotone submodular objective, which models diminishing returns, an algorithm maintains a set of at most $k$ available elements and changes only $O(1)$ elements after each insertion. Dütting et al. [2025] established a tight $2/3$ approximation with unrestricted computation and a polynomial-time $0.51$ approximation. They left open at STOC 2025 whether efficient algorithms can match the offline $1-1/e$ guarantee. We resolve this problem by proving that the supremum approximation achievable with polynomially many value queries and worst-case constant recourse is \[ β=2-\sqrt2\approx0.5858<1-1/e. \] For every $\varepsilon>0$, our randomized algorithm attains $β-\varepsilon$ with $O(\varepsilon^{-2})$ changes per insertion. Any fixed improvement requires exponentially many queries before one critical insertion or linear recourse of $Ω(k)$ changes at that insertion, even with unlimited queries afterwards. This gap quantifies the cost of consistency: the current oracle hides which elements will be needed after an arrival. We also determine the exact curvature-dependent threshold $1-(\sqrt2-1)\vartheta$, attain $1-1/e-\varepsilon$ for weighted coverage with $O(\varepsilon^{-1})$ recourse, and separate the existence of universal future-price certificates from their efficient computation. Our algorithm has a bounded-bit polynomial-time implementation for polynomial-bit rational oracle answers; the lower bound uses only logarithmic-bit rational answers.

Thursday, September 10 2026, 00:00

Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes

Authors: Akira Kitaoka

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of $\log T$ between the regret upper bound $O(d \log T)$ and the lower bound $Ω(d)$ is unresolved (here $T$ is the total number of rounds and $d$ is the dimension). When the action set is M-convex, the regret is known to be bounded by $O(d \log d)$, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake in the case where the correct action is uniformly separated from the other candidates, and applies it to online gradient descent, the online Newton step, and MetaGrad. The number of mistakes is then bounded, for all three, by a quantity independent of $T$; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes $O(d^2)$ when the forward problem is an integer linear program, that is, the factor $\log T$ is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.

Authors: Akira Kitaoka

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of $\log T$ between the regret upper bound $O(d \log T)$ and the lower bound $Ω(d)$ is unresolved (here $T$ is the total number of rounds and $d$ is the dimension). When the action set is M-convex, the regret is known to be bounded by $O(d \log d)$, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake in the case where the correct action is uniformly separated from the other candidates, and applies it to online gradient descent, the online Newton step, and MetaGrad. The number of mistakes is then bounded, for all three, by a quantity independent of $T$; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes $O(d^2)$ when the forward problem is an integer linear program, that is, the factor $\log T$ is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.

Thursday, September 10 2026, 00:00

Fast Algorithms for Sparse PCA and Robust Sparse Estimation

Authors: Giannis Iakovidis, Ankit Pensia

We study fast algorithms for sparse-PCA certification. Given a positive semidefinite matrix $M$, the problem asks either to rule out a large $k$-sparse quadratic form or to return a high-value (relaxed) witness. The standard semidefinite relaxation provides such certificates, but existing general-purpose solvers require $Ω(d^4)$ time. We give a bicriteria algorithm running in $O(d^2+d k^{O(\log k)})$ time: if some $k$-sparse unit vector has quadratic form greater than $2$, it returns either an $O(k^2)$-sparse unit vector or an SDP-feasible matrix of value at least $1$. For $k\leq\exp(O(\sqrt{\log d}))$, this running time is $O(d^2)$. We also go below the quadratic barrier in the sample-access model: Given $n=d^{o(1)}$ samples, our algorithm obtains a related one-sided certificate in $d^{2 - Ω(1)}$ time for $k=\mathrm{polylog}(d)$, without forming the empirical covariance matrix. As an application, these certificate routines yield the first quadratic and subquadratic-time algorithms for robust sparse estimation for broad families of distributions. Our sparse-PCA algorithm reduces a high-value sparse direction to a bounded-radius set in the graph of large correlations and searches the resulting candidate supports. The subquadratic implementation constructs this graph using fast correlation detection.

Authors: Giannis Iakovidis, Ankit Pensia

We study fast algorithms for sparse-PCA certification. Given a positive semidefinite matrix $M$, the problem asks either to rule out a large $k$-sparse quadratic form or to return a high-value (relaxed) witness. The standard semidefinite relaxation provides such certificates, but existing general-purpose solvers require $Ω(d^4)$ time. We give a bicriteria algorithm running in $O(d^2+d k^{O(\log k)})$ time: if some $k$-sparse unit vector has quadratic form greater than $2$, it returns either an $O(k^2)$-sparse unit vector or an SDP-feasible matrix of value at least $1$. For $k\leq\exp(O(\sqrt{\log d}))$, this running time is $O(d^2)$. We also go below the quadratic barrier in the sample-access model: Given $n=d^{o(1)}$ samples, our algorithm obtains a related one-sided certificate in $d^{2 - Ω(1)}$ time for $k=\mathrm{polylog}(d)$, without forming the empirical covariance matrix. As an application, these certificate routines yield the first quadratic and subquadratic-time algorithms for robust sparse estimation for broad families of distributions. Our sparse-PCA algorithm reduces a high-value sparse direction to a bounded-radius set in the graph of large correlations and searches the resulting candidate supports. The subquadratic implementation constructs this graph using fast correlation detection.

Thursday, September 10 2026, 00:00

Scalable Composition of Byzantine Agreements under Reorder Attacks