Theory of Computing Report

Authors: Azem Adibelli, István Tomon

We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tildeΩ(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.

Authors: Sidhant Saraogi

Matching Vector families (MVFs) are defined by two ordered lists of vectors in $\mathbb{Z}_m^n$ whose inner products satisfy specific residue patterns modulo an integer $m$. Most famously, restricted MVFs are used to construct the best-known constant-query Locally Decodable codes (LDCs). We prove an upper bound of $2^{O\left(\sqrt{n\log n \log m}\right)}$ on the size of $3$-restricted MVFs in $\mathbb{Z}_m^n$ for $m \leq \sqrt{n}$, substantially improving on the previous best bound of $2^{O(n/\log n)}$ by Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14). Our proof relies on a new polynomial method argument that controls collisions in sumsets of matching vectors.

Authors: Manuel Fernandez

We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.

Authors: Pedro M. M. de Castro

We consider an online geometric connection rule that stores one point of state. After processing $p_i$, the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$, and the two segments from $x_{i-1}$ to $x_i$ and from $x_i$ to $p_i$ are retained. The design parameter $γ$ controls the persistence of the state. We minimize the sum of the $α$-powers of the resulting edge lengths under independent uniform input and under arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $α=1$. Its continuous extension is minimized at the boundary for $0<α\leq1$, and every global minimizer is interior for $α>1$. The main result determines the finite optimizer in the joint window $α_N=1+\varepsilon_N$, $\varepsilon_N\log N\toλ$. Below an explicit threshold it lies on the $N^{-1/2}$ scale. At the threshold its scale is $\sqrt{\log N/(N\log\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies which correction governs the location, and differentiated estimates prove eventual uniqueness. At $α=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. Under arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per processed point are explicit for $0<α\leq3$. At high powers, periodic blocks and a separation argument show that this optimized cost is asymptotic to $2\log2/\logα$. Exact results for powers two and four, a rational recursion for all even powers, and a high-dimensional expansion provide additional descriptions of the optimizer.

Authors: Gabriel de Azevedo, Ulysse Hennebelle

In ranked-choice voting, a Condorcet-winning set is a group of candidates for which no outside candidate is preferred to every member of the group by a majority of voters. We study Condorcet-winning sets in planar metric elections, where voters rank candidates according to their distance under a given norm. We formulate general metric elections, peer selection, and the one-round Voronoi game as instances of a two-player Stackelberg game and place these problems in a common hierarchy. We also introduce a new variant, which we call strong peer selection. Our main result concerns peer selection under the $\ell_1$ and $\ell_\infty$ norms. We prove that every planar instance admits a Condorcet-winning set of size at most three, even under strong peer selection. This follows from a new result for strong rectangular $\varepsilon$-nets. We show that, for every set of points in the plane, one can choose at most three input points that intersect every axis-parallel rectangle containing more than half of the points, improving the previous threshold of $9 / 16$ due to Ashok et al. Under the $\ell_2$ norm, we prove that every planar metric election admits a Condorcet-winning set of size at most four, improving on the general bound of five due to Song et al. Finally, we give new norm-independent bounds for the one-round Voronoi game.

Authors: Michael A. Bender, Alex Conway, Martín Farach-Colton, Hanna Komlós, William Kuszmaul, Nicole Wein

The classical memory-allocation problem captures the task of placing objects of different sizes in memory, while minimizing the so-called memory high-water mark. It has been known since the early 1970s that the optimal competitive ratio for any deterministic online allocator is $Θ(\log M)$, where $M$ is the volume high-water mark of the underlying request sequence. This paper begins with a simple observation: many real-world allocators seem to bypass the 1971 lower bound by adopting a slightly different model for memory allocation. These allocators use what we call $k$-aggregate request fragmentation, meaning that the memory allocator is permitted to break requests into multiple fragments, so long as the all-time maximum number of simultaneous fragments is at most $k$ times the all-time maximum number of simultaneous requests. We consider the following basic question: Does request fragmentation fundamentally change the problem of memory allocation, and if so, how? Our results come with several surprises. Among these, we find that even using $k = 1 + o(1)$ request fragmentation, the optimal competitive ratio---which was $Θ(\log M)$ in the classical setting---collapses to $Θ(\log \log M)$. This result is shown to be tight with matching upper and lower bounds, applying to both deterministic and randomized algorithms.

Authors: Nikhil Bansal, Haotian Jiang

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.

Authors: Karthekeyan Chandrasekaran, Raymond Jiang, Krishna Kalathur

Motivated by non-uniform edge failures in network design, we introduce a multi-tier model of flexible graph connectivity. In k-tier Flexible Graph Connectivity (k-tier FGC), the input is an undirected graph G=(V, E) with non-negative edge costs, along with a classification of the edges into nested tiers T_1 subseteq T_2 subseteq ... subseteq T_k = E and non-negative integral tier requirements q_1 <= q_2 <= ... <= q_k. A non-empty proper subset R of vertices is safe if it is safe along one of the tiers, i.e., there exists i in [k] such that |delta(R) cap T_i| >= q_i. The goal is to find a minimum cost subset F subseteq E of edges such that the subgraph (V, F) has no unsafe cuts. The case of k=1 corresponds to the min-cost p-edge-connected spanning subgraph problem which is APX-hard. We design approximation algorithms for every fixed constant k for three variants of k-tier FGC: (i) for k-tier FGC, we design an LP-based logarithmic approximation, (ii) for min-cardinality k-tier FGC, we design a combinatorial approximation whose factor depends only on the tier requirements q_1 and q_k, and (iii) for k-tier Flexible Multi-Graph Connectivity, where we are allowed to use multiple copies of each edge while paying the cost of the edge for each chosen copy of the edge, we design an LP-based 2-approximation.

Authors: Lorenzo Beretta, Cameron Musco

We study random sketching matrices with Khatri-Rao structure. In particular, we consider the Khatri-Rao product (i.e., column-wise tensor product) $A_1\odot\cdots\odot A_d \in \mathbb R^{(n_1 \cdots n_d) \times m}$ of random matrices $A_i \in \mathbb R^{n_i \times m}$ whose columns are isotropic, independent and sub-Gaussian (e.g., Gaussian matrices). Khatri-Rao sketching matrices are widely applied in randomized algorithms for linear algebraic computation and data analysis, when the input data has tensor structure that allows for fast multiplication with $A_1\odot\cdots\odot A_d$. However, existing theory is not able to fully explain their performance in practice. In particular, despite significant attention, our best bounds for the important \emph{oblivious subspace embedding} property with Khatri-Rao matrices lag behind what is achievable with standard unstructured matrices. For embedding a $k$-dimensional subspace to $(1\pm ε)$ error, Bujanović et al. \cite{bujanovic2025subspace} prove that sketching dimension $m = O(k^{3/2}/ε^2)$ suffices in the special case of $d = 2$. Their dependence on $k$ is weaker than the tight bound of $O(k/ε^2)$ known for unstructured sub-Gaussian sketching matrices. In this work, we close this gap, showing that $m = \tilde O(k/ε^2)$ suffices for subspace embedding with a Khatri-Rao sketching matrix with any fixed order $d$. Our proof is simple, leveraging just two basic properties of the Khatri-Rao sketching distribution: 1) the columns of $A_1\odot\cdots\odot A_d \in \mathbb R^{(n_1 \cdots n_d) \times m}$ are independent and isotropic, and 2) each column of $A_1\odot\cdots\odot A_d \in \mathbb R^{(n_1 \cdots n_d) \times m}$ satisfies a weak Johnson-Lindenstrauss type moment property.

Authors: Kushagra Chatterjee, Rojin Rezvan, Ali Vakilian

We study the \emph{connected $k$-median} clustering problem, a clustering problem that augments the classical $k$-median objective with connectivity constraints. We focus on the \emph{overlapping} variant of the problem, where clusters are allowed to share vertices. In addition to a metric space $(V,d)$, the input contains a connected graph $G$ on the same vertex set $V$ of size $n$. The goal is to select at most $k$ centers $C$ and assign vertices to them so as to minimize the $k$-median cost (i.e., $\sum_{v\in V} d(v,C)$), subject to the constraint that each cluster induces a connected subgraph of $G$. Since the metric space and the connectivity graph are independent, the problem is significantly more challenging than standard clustering. Eube et al.~\cite{eube2025esa} showed that even the assignment version is $Ω(\log n)$-hard to approximate and gave approximation algorithms with guarantees depending polynomially on $k$. We develop a configuration-LP-based framework that combines covering LP techniques with a rooted minimum-density oracle. For the assignment version, we obtain an $O(\log^2 n)$-approximation. For the general version, we develop a bicriteria framework that opens $O(k\log n)$ centers while achieving an $O(\log^2 n)$-approximation in cost. %Our results provide a different LP-based approach for handling connectivity constraints in clustering problems and demonstrate that configuration LPs, covering LPs, and rooted density oracles can be combined effectively to obtain approximation guarantees for clustering objectives under graph-theoretic constraints.

Authors: Nima Anari

The canonical Bethe approximation gives a deterministic approximation to the permanent of every nonnegative matrix within a factor of $(\sqrt{2})^n$. We improve the base of this exponential factor: for some absolute constant $c<\sqrt{2}$, there is a deterministic polynomial-time $c^n$-approximation for the permanent of every nonnegative matrix. This shows that the canonical Bethe guarantee is not a barrier for deterministic approximation of the permanent. The proof augments the Bethe lower bound with a new certificate tailored to matrices on which that lower bound loses nearly the full factor. The author supplied the high-level plan of attack, and the proof was developed in an interaction with ChatGPT 5.6 Sol Pro. The author subsequently verified the results. Codex assisted with proof checking, manuscript assembly, and typesetting.

Authors: Jiaheng Cheng, Agastya Vibhuti Jha, Haotian Jiang

Numerical integration---approximating the integral of a function $f$ using $n$ point evaluations---is a central task in science and engineering. The two main paradigms for this problem, the Monte Carlo and quasi-Monte Carlo methods, have distinct strengths and limitations, and a fundamental question is to design a method that combines the benefits of both. \smallskip Building on recent algorithmic advances in discrepancy theory, Bansal and Jiang \cite{BJ25a} gave a randomized QMC method that naturally bridges the MC and QMC error guarantees. Their method also achieves a surprising improvement over the classical Koksma--Hlawka inequality for QMC methods: it attains an error bound of $\widetilde{O}(σ_{\mathsf{SO}}(f)/n)$, where $σ_{\mathsf{SO}}(f)$ is a new notion of \emph{smoothed-out variation} that they introduced and showed to be substantially smaller than the Hardy--Krause variation governing the classical bound. \smallskip However, the analysis in \cite{BJ25a} is quite involved: it must carefully exploit the structure of the dyadic decomposition and the randomness of the algorithm inside a sufficiently fine discretization of the Hlawka--Zaremba formula to obtain cancellations among the high-frequency components in the Fourier decomposition of $f$. The contribution of this article is twofold: (1) We give an equivalent characterization of $σ_{\mathsf{SO}}(f)$ in terms of the Haar--Besov seminorm of $f$, relating this new notion of smoothed-out variation to classical quantities. (2) Through this characterization, we provide a conceptually simpler and more direct analysis of the Bansal--Jiang QMC method via Haar decomposition, bypassing the use of the Hlawka--Zaremba formula, Fourier decomposition, and the delicate cancellation arguments of \cite{BJ25a} that heavily exploit the structure of dyadic decomposition.

Authors: Edith Cohen, Vadym Doroshenko, Badih Ghazi, Pritish Kamath, Alexander Knop, Ravi Kumar, Ethan Leeman, Pasin Manurangsi, Adam Sealfon, Marika Swanberg

We study differentially private (DP) $k$-means and $k$-median clustering in the online streaming setting. In this model, points arrive sequentially, and at each time step, we need to output a set of $k$ centers that optimizes the clustering objective for all points seen so far. We give a generic reduction that transforms the (sensitive) input stream into a private stream, which is a semi-coreset of the input stream. This implies that any (non-private) online clustering algorithm, run as a post-processing step, can achieve good utility for the original clustering objective. Our algorithm matches or improves upon the approximation ratio, space usage, and running time of existing algorithms [Epasto et al., 2026, Dupré la Tour et al., 2024]. A key aspect of our reduction is that it inherits desirable properties of the underlying non-private clustering algorithm, such as consistency [Lattanzi and Vassilvitskii, 2017]--a property not satisfied by previous DP algorithms.

Authors: Jiale Chen

We study dynamic maximum weight matching (MWM) under edge insertions and deletions in two settings: maintaining a $(1\pm\varepsilon)$-approximation to the optimum weight, and maintaining an explicit $(1-\varepsilon)$-approximate matching. Our main result is a reduction that transforms instances of polynomial aspect ratio into instances of aspect ratio $O(\varepsilon^{-1})$. The reduction applies to general graphs in both settings and is compatible with partially dynamic updates. The reduction is based on a structural property of local marginals. After grouping edges into weight classes, the global marginal contribution of one class relative to all lower classes is approximated by its marginal contribution within a local weight window of aspect ratio $O(\varepsilon^{-1})$. Summing these local marginals yields a value composition lemma that uses only approximate optimum values of the local windows. This improves the value reduction of Gupta and Peng (FOCS 2013), whose local aspect ratio is $\varepsilon^{-Θ(\varepsilon^{-1})}$. The same structural property yields an improved matching composition lemma for explicit matchings, reducing the local aspect ratio of Bernstein--Chen--Dudeja--Langley--Sidford--Tu (SODA 2025) from $O(\varepsilon^{-2})$ to $O(\varepsilon^{-1})$.

Authors: Navid Eslami, Ioana O. Bercea, Niv Dayan

Range filters are compact probabilistic data structures that answer approximate range emptiness queries. They are used in many domains, e.g., in key-value stores, to quickly rule out the existence of keys in a given query range and avoid searching for them in storage. However, all existing range filters exhibit at least one of three shortcomings: (1) they do not provide any false positive rate or performance guarantees, (2) they do not support variable-length keys and query ranges, and (3) they do not allow dynamic updates. We introduce Diva, the first range filter to address all the above challenges simultaneously. Diva learns the dataset's distribution by sampling keys and storing them in a cache-efficient trie. It compresses keys in-between samples by removing their longest common prefix and truncating their suffixes while leaving enough bits in the middle (i.e., an infix) to differentiate the keys in sorted order. It stores infixes in constant-time dynamic data blocks, which it splits to handle insertions and expansions. It processes a range query by traversing the trie and checking for the inclusion of infixes in the target query range. We mathematically prove that Diva provides the best possible trade-off between memory and false positive rate on many common real-world data distributions. We extend these benefits to a wider range of real-world workloads by introducing Diva++, an enhanced Diva variant. Diva++ saves memory by removing redundancies among infixes using order-preserving entropy encoding. It then removes any remaining identical infixes and uses the freed space to store more bits of the original keys within compact binary tries. We compare Diva and Diva++ to all prior range filters, and show that they achieve a false positive rate on par with the state of the art on real-world datasets while supporting dynamicity and variable-length queries and keys.

Authors: Hong Li

We give an approximation algorithm for the rural postman problem with approximation ratio strictly smaller than $3/2$. We obtain this result by adapting to the rural postman problem the technique of sampling from maximum entropy distributions for the metric traveling salesman problem of Karlin, Klein, and Oveis Gharan. We also observe that, for every fixed $\varepsilon>0$, any $α$-approximation algorithm for the metric traveling salesman problem yields an $(α+\varepsilon)$-approximation algorithm for the rural postman problem; this implication is already implicit in the treatment of edges that must be traversed in the work of Lampis on the inapproximability of the traveling salesman problem.

Authors: Karl Bringmann, Ariel Kulik, Karol Węgrzycki

We revisit the $d$-dimensional Vector Knapsack problem ($d$-Knapsack): Given a $d$-dimensional capacity vector and a set of items, each with a $d$-dimensional weight vector and a profit, the goal is to select a set of items that maximizes the total profit without exceeding the capacity in any dimension. For any $d\ge2$, the best known approximation scheme for $d$-Knapsack runs in time $O(n^{\lceil d/\varepsilon\rceil-d})$ [Caprara, Kellerer, Pferschy, Pisinger '00]. We improve this running time to $\widetilde O_{d,\varepsilon,ρ}(n^{\lceil\frac{d-1}{2\varepsilon}-\frac12+ρ\rceil}+n^d)$ for any $\varepsilon\in(0,1)$ and every parameter $ρ\in(0,1)$. We achieve this speedup by designing the first meet-in-the-middle algorithm for $d$-Knapsack. This requires replacing the LP solver used in prior algorithms by a highly efficient dynamic programming algorithm to generate representative solutions, building on an LP-based structural argument. This is the first improvement in over 25 years, and the first result that improves the exponent by a constant factor. We complement this by a fine-grained lower bound based on $k$-SUM showing that 2-Knapsack requires time $n^{\lceil\frac1{2\varepsilon}-\frac12 \rceil-o(1)}$. This establishes the optimal exponent of 2-Knapsack as $\frac1{2\varepsilon}\pm O(1)$, which is precise up to an additive $O(1)$. To the best of our knowledge, this is the first result that determines the optimal exponent more precisely than up to a factor $O(1)$, for any problem that admits a PTAS but no EPTAS. For the special case of 2-Knapsack we further attain a $(1-\varepsilon-δ)$-approximation in time $\widetilde O_{δ,\varepsilon}(n^{\lceil\frac1{2\varepsilon}-\frac12\rceil})$. This nearly matches our lower bound, as for a slightly better approximation ratio a slightly better running time is impossible -- so our algorithm is bicriteria-optimal.

The recent paper \cite{chatterjee2026bipartite} showed that deciding whether a bipartite graph has a perfect matching can be reduced to deciding whether a determinant, whose value may be assigned to any sufficiently large field $\mathbb{F}$, equals to zero. In the second part of their work, \cite{chatterjee2026bipartite} also generalized the algebraic method and proposed an $\mathsf{NC}$ algorithm that computes the maximum \emph{weighted} perfect matching of bipartite graphs. However, the method they used relies on the positivity of non-zero sum of squares and does not generalize to finite fields. In this paper we further leverage their ideas and show that the algebraic method works in finite fields as well, therefore maximum weighted perfect bipartite matching is in $\text{Mod}_p \mathsf{L}$.

In this note, we introduce the notion of ranked spreadness, a strengthening of the usual spread condition in which the elements of each member can be ordered so that their one-coordinate marginals decay geometrically with their rank. This additional structure removes the dependence on the maximum set size in random-containment estimates. We prove width-free hitting and weighted-concentration theorems for ranked-spread set systems, together with an elementary kernel-extraction theorem showing that ranked spreadness arises naturally in arbitrary distributions on small sets. Our main application is to the simulation of nonadaptive property testers by sample-based testers. If a one-sided tester has average query complexity $d$ and rejects every far input with probability at least $\delta$, then, for every integer $c>d/\delta$, it admits a one-sided sample-based simulation with expected sample complexity $O_{d,\delta,|\Sigma|}\bigl(n^{1-1/c}\bigr)$. More generally, if positive inputs are rejected with probability at most $\gamma$ and far inputs with probability at least $\delta>\gamma$,the same conclusion holds for every $c>d/(\delta-\gamma)$. In particular, for constant-query nonadaptive testers we obtain an exponent $1-\Theta(1/q)$, matching, up to the dependence on the rejection gap, the exponent conjectured by Fischer, Lachish, and Vasudev.

It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford $C^{r-1}Z$ gate, for every constant $r\geq 3$. In particular, we obtain length-$n$ quantum LDPC codes (with constant-weight stabilizers) of polynomial distance $d\geq n^{(1-\epsilon)/r}$ supporting transversal $C^{r-1}Z$ gates on a close-to-linear number $k\geq n^{1-\epsilon}$ of disjoint tuples of logical qubits, for arbitrarily small $\epsilon>0$. Our construction is the first with constant-weight stabilizers that obtains $dk\gg n$, and as a consequence achieves arbitrarily small magic state overhead exponent $\gamma=\log(n/k)/\log(d)>0$. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many $k=\Omega(n)$ logical $C^{r-1}Z$ gates, though with stabilizer weight and physical circuit depth $n^\epsilon$. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal $C^{r-1}Z$. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.

Consider the following problem of learning an unknown linear order on $n$ items. In each round, the learner guesses a complete ordering of the items and receives either confirmation that the guess is correct or a counterexample: a pair of items in the wrong order. The goal is to identify the unknown order using as few queries as possible. We study this problem when up to $k$ of the returned counterexamples may be untruthful, where $k$ is not known in advance. We determine the optimal query complexity up to constant factors: \[ \Theta(n\log n + nk). \] Thus, while the noiseless complexity matches the classical complexity of sorting, each untruthful counterexample incurs an additional cost of order $n$. The upper bound is based on a geometric representation of permutations and Gr\"unbaum's theorem, while the lower bound combines sorting arguments with a Condorcet-type construction. We also study the case where the target ranking has a low-dimensional geometric representation: each item is represented by a point in $\mathbb{R}^d$, and the ranking is obtained by projecting the points onto an unknown direction. For these classes we give an upper bound of $O(d^2\log n+dk)$ and a lower bound of $\Omega(d\log n+dk)$, leaving a factor of $d$ gap in the noiseless term.

We prove exponential lower bounds for $k$-DNF resolution on random $3$-CNF formulas throughout the range $k=O(\sqrt{\log n})$ at every constant clause density above the elementary first-moment bound for unsatisfiability. For random $3$-CNFs this improves Alekhnovich's range $k=O(\sqrt{\log n/\log\log n})$, and matches the $O(\sqrt{\log n})$ range obtained by Sofronova and Sokolov for random CNFs of sufficiently large constant width. We also obtain higher-density tradeoffs. In particular, random $3$-CNFs with $n\log^h n$ clauses are exponentially hard for $k=O(\sqrt{\log n/\log\log n})$ for every fixed $h>0$, while for every fixed $K$ the same conclusion holds simultaneously for all $1\le k\le K$ with $n^{1+\varepsilon}$ clauses whenever $\varepsilon<1/(4K^2+2)$.

In the Shortest Common Superstring problem (SCS), one is given a finite set of strings and is asked to find a shortest string containing every input string as a substring. Its best known approximation ratio is $2.466$, whereas the currently strongest upper bound on the approximation guarantee of the maximum-overlap greedy algorithm is $3.396$ (Englert, Matsakis, and Vesel{\'y}, 2023), though it is conjectured to be 2. We improve both approximation guarantees: SCS admits a $\frac{7}{3}$-approximation and the approximation guarantee of the greedy algorithm is at most $3$. The main technical ingredient of our proof is a certain inequality for optimum cycle covers of an overlap graph associated with the input strings. Every previous improvement of greedy's worst-case guarantee and the two recent record guarantees for general SCS are driven by it. We improve this inequality by pushing it to its limit: for a particular coefficient of this inequality, we show a new upper bound and prove that it cannot be improved further.

Matching Vector families (MVFs) are defined by two ordered lists of vectors in $\mathbb{Z}_m^n$ whose inner products satisfy specific residue patterns modulo an integer $m$. Most famously, restricted MVFs are used to construct the best-known constant-query Locally Decodable codes (LDCs). We prove an upper bound of $2^{O\left(\sqrt{n\log n \log m}\right)}$ on the size of $3$-restricted MVFs in $\mathbb{Z}_m^n$ for $m \leq \sqrt{n}$, substantially improving on the previous best bound of $2^{O(n/\log n)}$ by Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14). Our proof relies on a new polynomial-method argument that controls collisions in sumsets of matching vectors.

We prove an exponential separation between zero-sided-error randomized and two-sided-error pseudodeterministic communication complexities of partial boolean functions. This qualitatively improves and simplifies the proof of the separation by Göös, Harms, Riazanov, Sofronova, Sokolov, and Yuan (STOC 2026), which had a two-sided-error randomized upper bound. Then, we generalize our technique to separate the communication complexity analogues of MA and NP^BPP.

We show that a (stronger) version of random resolution has the feasible disjunction property. This is the first instance of a proof system not known to have feasible interpolation, which nevertheless has the feasible disjunction property.

Authors: Martin Kassabov, J. M. Landsberg, Victor Souza, Philip Speegle

This paper addresses centroids, which are fundamental invariants of tensors. Our main results are as follows: (i) The construction of explicit tensors with very large centroids, whereas previously it had been conjectured that none such exist. (ii) An upper bound on the dimension of the centroid that is essentially attained by our examples. (iii) The development of a geometric technique to write down border rank decomposition of tensors using centroids and "extended centroids". (iv) The technique is applied to tensors of this paper to prove they are of minimal border rank. The technique is versatile and enables us to geometrically derive and improve upon previous ad hoc decompositions. (v) The construction of symmetric tensors with large centroids and proof that they are wild in the sense of Buczyńska-Buczyński. Our results also pave the way for new upper bounds on the exponent of matrix multiplication. The geometric technique also constructs new "better" tensors for Strassen's laser method from old, and we apply this to the tensors of Strassen and Schönhage to get better tensors in the sense that they give better upper bounds on the exponent than the original tensors.

Authors: Thomas Watson

We prove an exponential separation between zero-sided-error randomized and two-sided-error pseudodeterministic communication complexities of partial boolean functions. This qualitatively improves and simplifies the proof of the separation by Göös, Harms, Riazanov, Sofronova, Sokolov, and Yuan (STOC 2026), which had a two-sided-error randomized upper bound. Then, we generalize our technique to separate the communication complexity analogues of MA and NP^BPP.

Authors: Benedikt Kolbe, Tim Mayr

We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of $n$-point metric spaces where the growth of the doubling dimension is in $o(\log n)$ and the diameter is bounded. In the $1$-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive $\varepsilon$-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in $n$. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability. We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris--Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.

Authors: Ondřej Draganov, Herbert Edelsbrunner, Sophie Rosenmeier, Morteza Saghafian

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a \emph{lune}, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its \emph{cost} is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

Authors: Eunku Park

We study nearest and farthest Voronoi diagrams of lines in $\mathbb{R}^3$ under the Euclidean metric when all $n$ lines belong to one ruling of a smooth doubly ruled real quadric. For arbitrary line sites, the combinatorial complexity of the nearest Voronoi diagram is known only to lie between $Ω(n^2)$ and $O(n^{3+\varepsilon})$. Under general-position assumptions, we prove that both diagrams in the ruling class have at most $4n(n-3)$ vertices and $O(n^2)$ total combinatorial complexity. Conversely, for every $n \ge 4$, one ruling of a fixed non-rotational one-sheeted hyperboloid contains a general-position set of $n$ lines with at least $(n-2)(n-3)/2$ distinct regular nearest vertices, where regular means that exactly four lines support the vertex and their three defining bisectors meet transversely. Thus the worst-case complexity of the nearest Voronoi diagram in this class is $Θ(n^2)$, while the farthest diagram has $Θ(n^2)$ complexity for every general-position input, since it has exactly $n(n-1)$ three-dimensional cells. Under the Plücker embedding, the ruling is a conic, and the condition for a line to be tangent to a Euclidean sphere restricts to a binary quartic. At a regular vertex, the four supporting parameters exhaust its roots, and sign alternation forces two arcs of the parameter circle to be site-free. This leaves only $n(n-3)/2$ possible cyclic support types, while Bézout's theorem bounds the number of centers for each type by eight. The same reduction yields an exact $O(n^2)$-time algorithm that, after cyclically sorting the site parameters, enumerates all finite nearest and farthest vertices as constant-degree real univariate representations.

Authors: Joshua A. Grochow, Gülce Kardeş, Michael Levet

In this paper, we investigate the low-depth circuit complexity of Group Isomorphism in the multiplication (Cayley) table model. We prove the first circuit lower bounds for Group Isomorphism: namely, we show that every family of depth-$2$ Boolean circuits deciding Group Isomorphism requires quasipolynomial-size. We complement this with upper bounds of uniform depth-$2\frac{1}{2}$ circuits of quasipolynomial-size. A sequence of previous results from 1970-2025 progressively reduced the circuit depth from polynomial to $3\frac{1}{2}$; all of these results relied on the generator-enumerator strategy and, in fact, applied more generally to quasigroups. In contrast, our depth-$2\frac{1}{2}$ construction follows a fundamentally different strategy that exploits structure more specific to groups. We guess a composition series for each group, together with generators for its terms and the isomorphism types of its composition factors. We then inductively verify that the corresponding extensions at each level of the two composition series are compatible. A central part in this approach brings to bear the extensive work on the Short Presentation Conjecture, in tandem with the algorithmic theory of group extensions and cohomology.

Authors: Julian Teusch, Jörg Philipp Müller, Monika Sester

Operational requirements developed with the City of Braunschweig frame municipal micromobility planning under geofenced exclusions, mandatory retained sites, spacing rules, and area-level caps. Each policy edit requires a new feasible plan; full-set greedy takes tens of seconds per alternative at city scale. We present CLIPPER (Constraint-exact Low-latency Iterative Planning with Pooled Evaluation and Replay). It forms bounded candidate pools but recomputes exact current gains and checks every active constraint before selection. Coverage from each candidate alone sets the initial order. Offline full-set scans measure gains omitted by the pool; online, a conservative bound triggers expansion or audit. CLIPPER-F gives each proposal group the same number of candidate slots. Across Braunschweig, Munich, and Berlin, its mean coverage over complete chains stays within 0.245 percentage points of full-set greedy under the same policy, with 13.6--28.9 times lower mean rollout time. CLIPPER-A instead distributes one shared candidate budget across the groups. Under its coverage-prioritized policy, it uses 9--15% of full-set greedy's rollout time under the same policy, with mean gaps of 1.82 percentage points in Braunschweig, 0.12 in Munich, and 0.27 in Berlin. Together, CLIPPER enables rapid, replayable comparison of recorded city-scale planning states while enforcing every encoded model constraint.

Authors: Frederic Koehler, Youngtak Sohn

We establish a sharp phase transition for fitting random vectors by an ellipsoid. The random vectors have independent subgaussian coordinates with mean zero, variance one, and a common fourth moment, and the number of vectors is proportional to the square of the dimension. We identify an explicit satisfiability threshold such that, with high probability, a positive definite ellipsoid passes through every data point below the threshold, whereas no positive semidefinite fit exists above it. We also determine the optimal squared fitting error throughout the unsatisfiable regime. In particular, the threshold depends on the coordinate distributions only through their common fourth moment, revealing a fourth moment universality phenomenon. For standard Gaussian data the threshold is $1/4$, resolving the ellipsoid fitting conjecture.

Authors: Francesco Paolo Nerini, Francesco Bonchi, Arijit Khan, André Panisson

Correlation Clustering (CC) is a natural formulation of clustering in combinatorial optimization, which uses a graph representation of the input and does not require a pre-specified number of clusters. Given $n$ objects and a pairwise similarity function, the goal is to cluster the objects so that similar objects are put in the same cluster and dissimilar objects are put in different clusters. Despite its versatility, existing CC algorithms suffer from significant scalability issues and are inherently transductive: i.e., the algorithm must be executed from scratch for any new problem instance. In this work, we bridge this gap by leveraging Graph Neural Networks (GNNs) to solve Inductive Correlation Clustering, a novel generalization of the CC problem designed to handle unseen graph instances. By learning to exploit common structural patterns and node features during training, our framework generalizes to new graphs drawn from the same distribution with minimal computational overhead with respect to standard algorithms. We demonstrate the effectiveness and scalability of our approach through extensive experiments. Our framework not only excels in the inductive setting, e.g., lowering the inference time up to $5$ order of magnitude, while maintaining an approximation ratio within $~10\%$ of the best baseline solution, but also achieves competitive results on standard (transductive) CC benchmarks. Finally, we showcase a practical application of our framework as a learnable pooling mechanism for graph classification. Our results indicate that our method serves as an efficient pooling layer, enhancing the ability of GNNs to capture hierarchical structural information in networks.

Authors: Michael A. Bekos, Giordano Da Lozzo, Petr Hliněný, Michael Kaufmann

A track layout of a graph is a partition of its vertices into linearly ordered independent sets, called tracks, such that no two edges between the same pair of tracks cross. Given a graph, the goal in this context is to determine its track number, that is, the minimum number of tracks required for the graph to admit a track layout. In this work, we present upper bounds on the track number of graphs admitting a product structure. Our main contribution is an algorithm that computes a track layout with at most $(2h+1) \cdot r \cdot tn(H)$ tracks for every subgraph of the strong product $P^h \boxtimes K_r \boxtimes H$, where $P^h$ is the $h$-th power of a path $P$, $K_r$ is the complete graph on $r$ vertices, and $H$ is a graph with track number $tn(H)$. Combined with existing product-structure results from the literature, this algorithm yields upper bounds on the track number of several graph classes. For planar graphs, the obtained bound matches the current best-known upper bound of $225$. For $1$-planar and optimal $2$-planar graphs, our algorithm yields track layouts with at most $375$ tracks, while for genus-$k$, $k$-planar, $k$-framed, $k$-map, and $k$-string graphs it provides track layouts with a number of tracks that depends solely on $k$, thus establishing new upper bounds on the track number for these graph classes. The algorithm runs in linear time for planar graphs and, more generally, in $O(n + h \cdot r \cdot t + f_t(H))$ time whenever a corresponding product-structure decomposition of the input $n$-vertex graph is provided as part of the input, where $t=tn(H)$ and $f_t(H)$ is the time needed to compute a $t$-track layout of $H$. Furthermore, our algorithm only uses elementary linked-list data structures.

Authors: Luyao Fan, Jiayang Zou, Jiayang Gao, Jia Wang

We study the zero-error randomized query complexity of finding all minimal elements in an unknown $n$-element poset of width at most $w$. Previous work of Daskalakis, Karp, Mossel, Riesenfeld, and Verbin established a randomized upper bound with leading term $\frac{w+1}{2}n$, while the corresponding lower bound left a multiplicative gap in the leading constant that approaches a factor of 2 as $w$ grows. We prove the finite lower bound \( R^{\mathrm{LV}}_{n,w}\ge \frac{w+1}{2}n-\frac{w(w+3)}4 +w\left(1-\frac1w\right)^n +\frac{w(w-1)}4\left(1-\frac2w\right)^n. \) Consequently, for every fixed $w$, \( R^{\mathrm{LV}}_{n,w} = \left(\frac{w+1}{2}+o(1)\right)n. \) Thus the known randomized upper bound has the correct asymptotic leading constant for every fixed width. The argument is based on a pairwise accounting of incomparable queries under a random-chain hard distribution, using a component-flip involution and a unique ownership property for incomparable comparisons. Generative AI was used in the preparation of this manuscript.

Authors: Matic Požar

Recent work by Haeupler, Hladík, Rozhon, Tarjan, and Tětek on the instance optimality of shortest-path algorithms established several results concerning Dijkstra's algorithm and bidirectional Dijkstra's algorithm in weighted and unweighted graphs. Motivated by these results, we revisit the question of instance optimality for shortest $st$-path algorithms in the standard query model. We identify several issues in the analysis of the instance optimality of both unidirectional and bidirectional Dijkstra's algorithms and provide corresponding counterexamples. We then propose a minimal simple modification of the bidirectional Dijkstra algorithm and prove that the resulting variant is instance optimal in the weighted setting. Furthermore, we revisit the unweighted case, provide a simplified proof of the lower bound showing that no algorithm can achieve instance optimality up to a factor better than $O(Δ)$, where $Δ$ denotes the maximum degree of the graph, and discuss the implications of this result for approximation algorithms. Finally, we make progress on the open problem of instance optimality in simple graphs. We show that if the problem instance satisfies $n\ge m/16$, where $n$ is the number of nodes and $m$ is the number of edges queried by our algorithm, then it is optimal up to a constant factor. Additionally, we show instance optimality for a broad class of instances, in particular when the largest degree in the graph is at most the square root of the number of explored edges, our algorithm exhibits optimality up to a constant factor.

Authors: Omer Gurevich, Maor Matityahu, Tal Mor, Aryeh Lev Zabokritskiy

We revisit a degree-only arc Hamiltonian for fixed-fleet, homogeneous, uncapacitated vehicle routing. Because its local penalties define only a cycle cover, ground states may contain customer cycles disconnected from the depot. We construct a polynomial-size quadratic unconstrained binary optimization (QUBO) repair using capped single-commodity flow and prove that every ground-state routing is connected and cost-optimal under explicit penalty assumptions. For $N-1$ customers and $K$ nonempty routes, the unreduced encoding uses exactly $|E|(1+\lceil\log_2(N-K+1)\rceil)$ logical problem qubits. A reversible compute--phase--uncompute realization evaluates the flow penalties in $O(N^2\log N+N\log^2N)$ logical gates on a complete graph with $O(\log N)$ reusable workspace and no product register. On complete loopless graphs, a depot-delimited single-sequence position encoding uses fewer problem qubits and fewer written terms when the flow-word length grows. Conversely, the flow model achieves a smaller structured logical-gate upper bound under a common reversible accounting model. Exact audits of the Hamiltonian and circuit implementation, combined with a $1{,}200$-matrix classical benchmark, verify the formulation and quantify the connectivity gap. Finally, a 32,000-shot Amazon Braket task on IQM Emerald characterizes depth-one termwise Ising circuits on a diagnostic $N = 4,\, K = 1$ counterexample instance. In the degree-only circuit, $78.05\%$ of selected $p=1$ shots realize the invalid disconnected ground state; the reduced 14-qubit flow-augmented circuit yields no fully feasible sample. These device results characterize mapped Hamiltonians and compilation rather than an asymptotic routing solution advantage.

Authors: Xiaoyu Chen, Eric Vigoda, Xiongxin Yang

The permanent of an $n\times n$ $0/1$ matrix $A$ equals the number of perfect matchings in the bipartite graph with edges defined by $A$. Jerrum, Sinclair, and Vigoda (2004) presented an FPRAS for approximating the permanent of any nonnegative matrix using a novel simulated-annealing algorithm. The running time was improved by Bezáková, Štefankovič, Vazirani, and Vigoda (2008) to $O(n^7\log^4 n)$ for $0/1$ matrices, for any fixed approximation and success parameters. We present the first asymptotic improvement over this running time bound, obtaining an $O(n^6\log^5 n)$-time algorithm. As in the previous works, our algorithm extends to arbitrary nonnegative matrices. The analysis of Bezáková et al. yields an $O(n^4)$ relaxation time bound for the JSV Markov chain on perfect and near-perfect matchings with ideal hole weights, under which each hole pattern (the unmatched vertices, if any) is equally likely in the stationary distribution. We introduce a restricted Poincaré inequality for the partition into hole patterns and prove an $O(n^3)$ bound on the corresponding restricted relaxation time. Our proof uses a coupled multicommodity flow argument inspired by a recent transport-flow argument of Chen et al.~(2025) for the Jerrum-Sinclair chain on all matchings.

Authors: Zhao Song, Lichen Zhang

Randomized sketch-and-solve algorithms accelerate overconstrained $\ell_2$ regression by replacing the input with a smaller problem. Standard subspace embeddings guarantee that the cost of the regression is nearly preserved, but coordinate-wise accuracy of the solution is more delicate: we want the solution vector itself to be close to the optimal solution in $\ell_\infty$ norm. In particular, we want to find a vector $x'\in \mathbb{R}^d$ such that $\|x'-x^*\|_\infty\leq \fracε{\sqrt d}\cdot \|Ax^\star-b\|_2\cdot \|A^\dagger\|_{\rm op}$. Price, Song and Woodruff initiated the study of this problem and showed that the subsampled randomized Hadamard transform (SRHT) with $O(ε^{-2} d^{1+Θ(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee. A subsequent work of Song, Ye, Yin and Zhang claimed to improve the row count to $O(ε^{-2}d\log^3 n)$. Unfortunately, their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-$d$ row count, we introduce a new fast, dense randomized transform, which combines a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression in which the noise is independent of the entire sketched design; this conditional independence is exactly what the earlier argument was missing. Our sketch yields the $\ell_\infty$ guarantee with $m=O(ε^{-2}d\log d)$ rows, uses one Hadamard pass with a padded internal dimension $N=\widetilde{O}(n+ε^{-2}d^3)$, and is efficient to apply: the sketched pair $(SA, Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+ε^{-2}d^4)$ time.

Authors: Aaron Li, Yifan Li, Drew DeHaas, Giulia Guidi

Updating a graph by inserting or replacing nodes while preserving semantics and reusing existing structure is a recurring computational problem. In population genetics, this problem arises in the genotype representation graph (GRG), a directed acyclic graph that losslessly encodes phased genetic variation across hundreds of thousands of samples by sharing subgraph structure for individual mutations. In a GRG, each mutation's carrier set is implicitly encoded as the set of leaf nodes reachable from the node it is assigned to. Updating a mutation is therefore a structural editing problem, and current approaches remap mutations individually. This paper introduces a batched mutation-remapping algorithm that replaces independent reuse-aware traversals with a single shared reverse-topological pass, identifying reuse candidates for an entire batch at once. The pass propagates compact bit-parallel per-mutation state and uses an adaptive sparse/dense carrier set representation spanning rare-to-common variant densities. Batching is the memory-scalable complement to split-based parallelism, which instead replicates graph and traversal state per worker. Our remapping is evaluated on a controlled update workload and on end-to-end allele polarization, a bulk carrier set update that is common in population genetic analysis. Our approach is up to 10.5$\times$ faster than independent remapping while preserving exact carrier-set semantics.

Authors: Somya Nigam, Johan Springael, Kenneth Sörensen

The problem of finding a fair consensus ranking is an active research topic in the domain of fair rank aggregation and has been well studied; however, existing studies predominantly consider both the input rankings and the output ranking to be permutations where elements are always strictly ordered. In practice, however, a ranking with ties is far more common. This study bridges this gap by presenting a post-processing approach to determine the closest fair consensus ranking when an unfair tie-aware consensus ranking is provided. It proposes an exact algorithm and a fast heuristic to achieve this.

Authors: Francisco J. Soulignac

The time-dependent traveling salesman problem with time windows (TDTSPTW) generalizes the well-known traveling salesman problem with time windows by accounting the effects of congestion on travel times. In this paper, we develop an exact framework for the TDTSPTW with a makespan objective that extends the range of instances solvable to optimality under loose time windows while remaining effective across all levels of time-window tightness. Our framework relies on a dynamic-programming labeling algorithm and combines column generation, ng-memory augmentation, and exact search, using completion bounds for state-space sparsification, variable fixing, and exact search pruning. Embedded within a branch-and-price method, the framework solves all instances with up to 45 customers in a benchmark comprising more than 10,000 instances, including all instances without time windows with up to 50 customers.

Authors: Shamisa Nematollahi, Daniel Vaz

We consider the buy-at-bulk facility location problem (BBFL), a problem combining the classic facility location problem with buy-at-bulk network design, which finds motivation in telecommunication networks. In it, we are given a graph with edge lengths, opening costs and demands for each vertex, and a monotone and subadditive capacity-cost function, and our task is to open facilities on a subset of the vertices and route the demand from each vertex to these facilities. The cost of a solution (which we want to minimize) is given by the opening costs of the chosen facilities, plus the cost on each edge, which is given by its length times the cost of providing enough capacity for the demands through the edge, given by the capacity-cost function. A common variant of the problem, the $k$-cable facility location problem (kCFL), considers the case where capacity is provided by buying copies of given cable types, each with a certain capacity and cost. We study BBFL on tree instances and show, for the unit-demand and splittable variants, that the problem admits a PTAS (a $(1+ε)$-approximation for any $ε> 0$). We also consider kCFL in the new setting of cable-unsplittable demands, where the demand of a vertex cannot be split among multiple cables. We show that the problem is NP-hard to approximate to a factor better than $3/2$ on stars, and then provide an algorithm for tree instances that outputs a solution with optimal cost, but which exceeds the capacity on each cable by a factor of $1+ε$. As a consequence, we show that the problem has a $2$-approximation algorithm on trees.

I more or less said what I was going to do in my tongue-in-cheek, cryptic post on Tuesday, but let me dive into the full details of how I’m planning on structuring this grad seminar. I’m going to log the course content here, which now lists a rough schedule for the class sessions. Taking inspiration from Matt Jones and Chris Wiggins, I’m going to do a weekly split between culture and engineering. Focusing on a particular application domain each week, we’ll spend one session discussing the purpose of forecasts in that domain and the other on the methods.

I arranged things into a thematic arc, starting with the weather—forecasting’s biggest success story—then moving to shakier ground in seismology and epidemiology, and ending with, well, millenarianism. As we move across timescales, our ability to predict nature dissipates: we can make accurate ten-day forecasts, but predicting large-scale climate disruptions is far more qualitative. We can predict immediate earthquake impacts at a distance once an earthquake has happened, but we can’t nail down precisely when a big one will happen. What we do with precise, short-term forecasts is completely different from what we do with imprecise long-term forecasts. Short-term forecasts dictate actions; long-term forecasts of discrete shocks inform risk management, preparedness, and rapid-response policies. I’m hoping that by the end of the semester I can better articulate what long-term forecasts of the end of the world do.

We’ll look at forecasts in governance and how they influence and shape policy. I’m particularly interested in discussing the rise of cost-benefit analysis in US governance. Cost-benefit analysis puts a specific number on something completely unknowable, but is now mandatory for any bill or law to pass. I want to trace how we became so reliant on a particular set of methods for guessing costs and benefits. I’m also interested in how economists became convinced you could forecast “the economy.” This required inventing something called “the economy” that could be forecast in the first place. How our system of government got so tied to a particular style of economic prediction will occupy several weeks of the class.

We’ll also get into why we want to predict “the public.” I want to examine how opinion polls went from a question of legibility to one of prediction. How did we get obsessed with using surveys to predict outcomes like elections? In parallel, we’ll look at the history of attempts to simulate the public. Simulation is nice because you don’t have to talk to people, right? We’ll look at the many misses over the history of human simulation in policy scenarios, and dig into the current obsessions with using LLMs to predict what people might do.

Finally, we’ll get into the weird culture of competitive forecasting. We’ll engage with ideas from superforecasting and prediction markets and ask why people think these are useful information-processing systems. We’ll talk about punditry and how it’s not always interested in minimizing a Brier Score. We have to talk about the relationship between forecasts and gambling. And we’ll try to piece together why putting odds on outcomes makes people feel better about the future.

For the methods, I did my topic matching so you could extract a logically ordered half-semester course on forecasting from those lectures alone. This is not a class on how to be a rational forecaster. I want to problematize those methods more than tell you how to implement them. You can ask your friend Claude if you need an honest, load-bearing implementation.

Our methods survey starts with a refresher on my idiosyncratic views of machine learning as optimization-driven algorithmic pattern recognition. This will lead to a lot of discussion of the optimization problems themselves and why people like them. We’ll cover scoring rules, calibration, maximum likelihood, and utility maximization. We will discuss the role of models and look at probabilistic recurrence models, dynamical system models, differential equations, and other simulation-based tools. We’ll spend time on uncertainty quantification and how people come up with error bars (part of being a good forecaster is plausible deniability). Then we’ll look at offline and online optimization methods that let you fill in predictions based on your cost functions and modeling assumptions. I’m interested in highlighting the metrical determinism. The cost functions and models more or less tie your hands algorithmically, and most of the cleverness goes into how you evaluate.

Hopefully this arc will feel coherent as we go. I’m not into predictions, so don’t get mad if the story changes as I go. I’ll blog through it, and then we can reflect on where we land at the end of the semester.

Enrolled students (and those dedicated to following along at home) have an important first assignment: pick something to forecast. I don’t care what it is. Throughout the course, the goal is to learn the practical techniques by making predictions. Every week I’ll ask you to try to apply the tools to your problem. Or at least find how other people have applied those same tools to your problem. At the end of the semester, we’ll present our full findings and see how accurate we can be.

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By Ben Recht

Authors: Martin Koutecký, Nikolaos Melissinos, Tung Anh Vu, Lluís Sabater

Computational social choice seeks algorithmic answers to questions about preference aggregation, safety of elections, robustness of outcomes, stability, etc. It overwhelmingly models societies as composed of discrete agents. We propose to study computational social choice problems in a society continuum} setting, where a society is modeled as a distribution of infinitely many infinitesimal agents of different types. An analogous approach has been very useful in physics (it is the basis of statistical mechanics), economics (mean field games), and other fields. As an initial case study, we focus on election attacks (bribery and control), which have been extensively studied in the discrete setting. We show that a broad class of standard election attacks becomes polynomial-time solvable in the society continuum. The class contains problems that are NP-hard discretely, among them Borda- and Bucklin-CCDV and unit-cost Borda-SWAP BRIBERY. Furthermore, we give polynomial-time algorithms for $k$-Approval-SWAP BRIBERY when $k$ is constant for general costs, and when $k$ varies and the cost function is additively separable. The latter result contrasts with the discrete problem, which we prove NP-complete for additively separable costs and every fixed $k\ge 2$. In contrast, we prove that Borda-SWAP BRIBERY and $k$-Approval-SWAP BRIBERY, both with general costs, remain computationally hard in the society continuum. To obtain these results, we use both continuous and discrete optimization techniques, such as the Configuration LP framework and dynamic programming. Of particular note is the technique underlying our hardness proofs, which shows how to ''reverse the flow of hardness'' between LP formulations and pricing problems.

Authors: Youlong Ding, Aayush Jain, Ilan Komargodski

We present a new generic transformation from weak PRFs computable in depth $d(n) = Ω(\log n)$ to strong PRFs computable in depth $O(d(n))$. This construction refines the classical tree-based paradigm of GGM by {tapering} the internal state so the per-level depth decreases geometrically. We complement the above with new depth-efficient weak PRF constructions based on various standard assumptions. As a corollary, we obtain new $\mathsf{NC}^1$-computable PRFs from various classical assumptions, resolving several long-standing open problems. Concretely, for the first time, we obtain $\mathsf{NC}^1$-computable PRFs: (1) from the \textbf{Learning With Errors (LWE)} assumption with a polynomial modulus-to-noise ratio, improving upon prior low-depth constructions that required Ring-LWE with super-polynomial ratios [Banerjee-Peikert-Rosen, EUROCRYPT 2012]; (2)from the standard \textbf{Learning Parity with Noise (LPN)} assumption, removing the need for structured LPN variants [Boyle et al., FOCS 2020], [Ding-Jain-Komargodski, STOC 2025]; (3) from the \textbf{Computational Diffie-Hellman (CDH)} assumption; prior works relied on the stronger Decisional Diffie-Hellman (DDH) or generalized Diffie-Hellman (GDH) assumptions [Naor-Reingold, FOCS '97, J. ACM '04].

Authors: Pedro M. M. de Castro

Let $p_0,p_1,\ldots,p_N$ be points of the unit ball of $\mathbb R^d$, processed in a prescribed order. We study the insertion cost $\sum_{i=1}^N\lVert p_i-x_{i-1}\rVert^α$, where each $x_{i-1}$ is computed from the previously observed points. The input-order path is sensitive to the input distribution but can repeatedly pay the diameter under adversarial input. The center star has controlled worst-case scale but ignores the observed sequence. We compress the past into one point through $x_0=p_0$ and $x_i=γx_{i-1}+(1-γ)p_i$, where $0\leqγ\leq1$. Thus $x_i$ is an exponentially weighted memory of the input, maintained with one $d$-dimensional point of working state. For independent uniform points, the stationary insertion length is nonincreasing in the usual stochastic order as $γ$ increases. If $d\geq2$ and $α>0$, every optimal constant parameter for $N$ insertions satisfies $1-γ_N^*=Θ(N^{-1/2})$. We determine its asymptotic constant and the resulting $\sqrt N$ correction, with explicit bounds in $d$ and $α$. For $α=1$, the leading expected tree length equals that of the center star and is strictly smaller than those of the endpoint constructions. For $α=2$, the minimizer is unique for $N\geq2$, with $1-γ_N^*=N^{-1/2}-\tfrac12N^{-1}+O(N^{-3/2})$. For arbitrary input sequences and fixed $0\leqγ<1$, the largest asymptotic mean cost is $(2/(1+γ))^α$ for $0<α\leq3$, strictly below the path value when $γ>0$. Among fixed nonnegative weighting rules whose contributing points have the same average distance in the input order from the most recent point, exponential weighting is within a factor smaller than $1.161^α$ of the best adversarial value in dimension at least two; this ratio tends to one as that average distance grows.

Authors: Édouard Bonnet, Julien Duron, Marcin Pilipczuk, Marek Sokołowski, Szymon Toruńczyk

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.