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arXiv2026

Finding a Shortest Vector and More in $2^{n/2+o(n)}$ Time using $q$-ary Coset Difference Tree

This paper presents a new randomized algorithm for solving the exact shortest vector problem. For the $n$-dimensional lattice $\mathcal L$, our algorithm runs in time and space $2^{n/2+o(n)}$. Our algorithm can be viewed as a $q$-ary analogue of the midpoint Hessian for an odd prime $q$; more precisely, we use the fact that, for a shortest vector $v$, the gradient (rather than Hessian) of the periodic Gaussian function at $v/q$ is nearly proportional to $v$ (up to sign), even after aggregation over a relatively large random affine coset. We compute the relevant coset gradient along a chain of intermediate lattices using a combinatorial procedure inspired by Wagner's generalized birthday algorithm, yielding the $2^{n/2+o(n)}$ time and space complexity. A variant of the algorithm solves the exact closest vector problem on every input $(y,\mathcal L)$ with a distance guarantee $\operatorname{dist}(y,\mathcal L)\le 1.039λ_1(\mathcal L)$ within the same time and space complexity. This guarantee holds for a random target and a random lattice drawn according to the Haar-Siegel measure. Thus, this algorithm solves a closest vector problem on such random instances in time and space $2^{n/2+o(n)}$.

cs.DS↗

arXiv2026

Improved Multilayered PCPs and Hypergraph Vertex Cover

We present two elementary constructions of multilayered PCPs that improve upon prior constructions in two ways. Specifically, we give one construction of quasi-linear size, and another one with $2$-to-$2$ constraints. Using these constructions we obtain the following results for the hypergraph vertex cover problem: $\bullet$ For $k=3$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $3$-uniform hypergraph within factor $1+\sqrt{2}-\varepsilon$ is NP-hard. Previously, the best known result due to [Dinur, Guruswami, Khot, Regev, SICOMP 2005] achieved a factor of $2-\varepsilon$. $\bullet$ For $k\geq 4$, for all $\varepsilon>0$, approximating the minimum vertex cover of a given $k$-uniform hypergraph within factor $k-\varepsilon$ is NP-hard, which is tight. Previous works established this result assuming the Unique-Games Conjecture [Khot, Regev, JCSS 2008], and a weaker factor of $k-1-\varepsilon$ for standard NP-hardness [Dinur, Guruswami, Khot, Regev, SICOMP 2005]. $\bullet$ Assuming the Exponential Time Hypothesis, for all $k\geq 3$ and $\varepsilon>0$ there is $C>0$ such that no $2^{n/\log^C n}$-time algorithm approximates the minimum vertex cover in a $k$-uniform, $n$-vertex hypergraph within factor $k-1-\varepsilon$. The proofs were obtained using ChatGPT 5.6 Pro and subsequently rewritten by the communicators.

cs.CC↗

arXiv2026

Maximum Matching-Match: Hardness and Approximation

In this paper, we study \textsc{MaxMMP}, an optimization variant of the Matching-Match Puzzle introduced by Iburi and Uehara (FUN 2024). Given a graph, a partial vertex coloring, and a multiset of colored sticks, the goal is to complete the coloring and assign the sticks to graph edges so as to maximize the number of satisfied edges. We first prove that \textsc{MaxMMP} is APX-hard by an reduction from \textsc{Max-Cut}. The hardness already holds with two colors, no precolored vertices, and only bichromatic sticks. We then give a simple deterministic $\frac{2}{c(c+1)}$-approximation for completely uncolored instances, improving to $\frac{2}{c(c-1)}$ when all sticks are bichromatic. Next, we obtain a randomized $\frac{1-\frac{1}{e}}{2c}$-approximation for arbitrary instances with $c$ colors by reducing the remaining coloring choices to monotone submodular maximization under a partition matroid. On bipartite graphs, the approximation ratio improves to $\frac{1-\frac{1}{e}}{c}$. For every fixed $c$, we further obtain deterministic $\frac{1}{2c}$ and $\frac{1}{c}$-approximations on general and bipartite graphs, respectively, in time $n^{O(c^2)}$. Finally, for every fixed number of colors, we show that \textsc{MaxMMP} can be solved exactly in time $n^{O(c^2)}$ on trees and on cographs.

cs.DS↗

arXiv2026

Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph $F \in \mathcal{F}$. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems $\textrm{HomInd}(\mathcal{F})$ of deciding homomorphism indistinguishability over $\mathcal{F}$ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of $\textrm{HomInd}(\mathcal{F})$, Seppelt (MFCS 2024) showed that $\textrm{HomInd}(\mathcal{F})$ is in randomised polynomial time for every graph class $\mathcal{F}$ of bounded treewidth that can be defined in counting monadic second-order logic $\mathsf{CMSO}_2$. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in $\mathsf{PTIME}$. For $\mathsf{CMSO}_2$-definable graph classes $\mathcal{F}$ of bounded pathwidth, we improve the previous complexity upper bound for $\textrm{HomInd}(\mathcal{F})$ from $\mathsf{PTIME}$ to $\mathsf{C}_=\mathsf{L}$ and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as $\mathsf{C}_=\mathsf{L}$-complete.

cs.CC↗

arXiv2026

Near-Feasible Stable Matchings: Incentives and Optimality

Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works leave open deviation incentive issues that arise when the centralised authority modifies agents' capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.

cs.GT↗

arXiv2026

Tight Bounds for Memory Allocation With and Without Request Fragmentation

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.

cs.DS↗

arXiv2026

High-dimensional Linear Bandits with Knapsacks

We investigate the contextual bandits with knapsack (CBwK) problem in a high-dimensional linear setting, where the feature dimension can be very large. Our goal is to harness sparsity to obtain sharper regret guarantees. To this end, we first develop an online variant of the hard thresholding algorithm that performs the sparse estimation in an online manner. We then embed this estimator in a primal-dual scheme: every knapsack constraint is paired with a dual variable, which is updated by an online learning rule to keep the cumulative resource consumption within budget. This integrated approach achieves a two-phase sub-linear regret that scales only logarithmically with the feature dimension, improving on the polynomial dependency reported in prior work. Furthermore, we show that either of the following structural assumptions is sufficient for a sharper regret bound of $\tilde{O}(s_{0} \sqrt{T})$: (i) a diverse-covariate condition; and (ii) a margin condition. When both conditions hold simultaneously, we can further control the regret to $O(s_{0}^{2} \log(dT)\log T)$ by a dual resolving scheme. As a by-product, applying our framework to high-dimensional contextual bandits without knapsack constraints recovers the optimal regret rates in both the data-poor and data-rich regimes. Finally, numerical experiments confirm the empirical efficiency of our algorithms in high-dimensional settings.

cs.LG↗

arXiv2026

Scalable Neighborhood Local Search for Single-Machine Scheduling with Family Setup Times

In this work, we study the task of scheduling jobs on a single machine with sequence dependent family setup times under the goal of minimizing the makespan, that is, the completion time of the last job in the schedule. This notoriously NP-hard problem is highly relevant in practical productions and requires heuristics that provide good solutions quickly in order to deal with large instances. In this paper, we present a heuristic based on the approach of parameterized local search. That is, we aim to replace a given solution by a better solution having distance at most $k$ in a pre-defined distance measure. This is done multiple times in a hill-climbing manner, until a locally optimal solution is reached. We analyze the trade-off between the allowed distance $k$ and the algorithm's running time for four natural distance measures. Example of allowed operations for our considered distance measures are: swapping $k$ pairs of jobs in the sequence, or rearranging $k$ consecutive jobs. For two distance measures, we show that finding an improvement for given $k$ can be done in $f(k) \cdot n^{\mathcal{O}(1)}$ time, while such a running time for the other two distance measures is unlikely. We provide a preliminary experimental evaluation of our local search approaches.

cs.DS↗

arXiv2026

Strategic Facility Location in Euclidean Spaces

The strategic facility location problem is defined as follows: $n$ agents report their location in a metric space, and the objective is to design a \emph{mechanism} deciding the (possibly randomized) location of a facility such that agents have no incentive to lie about their position. We focus on the egalitarian cost, which means that the goal of the mechanism is to minimize the expected maximal facility-agent distance. Meanwhile, mechanisms must be \emph{truthful} (or \emph{strategyproof}): no agent may decrease their expected distance to the facility via lying on their location. Designing truthful mechanisms minimizing the approximation ratio is a well-studied problem, and the optimal solution is known for the real line. We focus in this paper on higher dimension Euclidean spaces, for which gaps remain between the best known lower and upper bounds. We first show that, maybe counter-intuitively, the problem is easier for two agents on the plane rather than on the line: the mechanism can exploit the additional dimension to prevent more efficiently agent lies. Based on this intuition, we devise lower bounds for $\mathbb R^d$ asymptotically matching the best known approximation factor of $2$ for large $d$. We also provide novel mechanism ideas, improving over the best known algorithms on the plane, and when the agents belong to $\mathbb R^d$ but the facility may use an additional dimension.

cs.GT↗

arXiv2026

Socially Fair Clustering: Parameterized Approximation and Local Search

We study the Socially Fair Clustering problem introduced by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021), along with its extension, the $(p,q)$-Socially Fair Clustering problem. This problem generalizes $k$-medians and $k$-means to settings where data points are partitioned into $\ell$ groups, and the goal is to find a fair clustering that is simultaneously good for all groups. We present several algorithms for this problem. For $\ell_p$-Socially Fair Clustering, we give the first constant-factor FPT-approximation parameterized by the number of groups $\ell$, resolving the open question raised by Ghadiri, Singh, and Vempala (2022). Our main ingredient is a new algorithm for closing additional centers in parameterized time inspired by local search. We then turn to the more general $(p,q)$-Socially Fair Clustering problem. The known algorithm for this problem, proposed by Chlamtáč, Makarychev, and Vakilian (2022) achieves a very good approximation but is complex, slow and difficult to implement. We analyze the performance of a simple local search algorithm and show that it provides an $O(q)$ approximation in the worst case. Finally, we design approximation algorithms for the facility location variant of the problem, where the number of facilities (centers) is not fixed in advance, and opening each facility incurs an opening cost. Unlike in previous work, we do not assume these opening costs are the same for all groups.

cs.DS↗

arXiv2026

A Tight Bound for Facial Distance Patterns in Planar Graphs

Let $G$ be an undirected unweighted planar graph and let $S=(s_0,\dots,s_{k-1})$ be the vertices of a designated face, listed in cyclic order. Consider a vector that stores the distances from an arbitrary vertex $v$ to all vertices of $S$. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values in this vector. Li and Parter [STOC'19] proved an upper bound of $O(k^3)$ on the number of unique patterns over all vertices of $G$. We improve this to $O(k^2)$, matching a known lower bound and settling a conjecture in [ISAAC'22]. The simple proof was found by OpenAI's GPT 5.6-Sol model. Plugging this new bound into known results has the following three immediate implications for undirected unweighted planar graphs: (1) it gives an improved compression of the Okamura-Seymour metric (2) it improves the space required by constant-time exact distance oracles, and (3) it improves the fastest distributed algorithm for computing the diameter. We further present a previously unknown and nontrivial implication: a (centralized) $\tilde{O}(n^{8/5})$-time algorithm for computing the diameter, improving over the $\tilde{O}(n^{5/3})$ algorithm of [SODA'18] which works for weighted directed planar graphs. Thus, there is currently a gap between the time for computing the diameter between weighted and unweighted planar graphs.

cs.DS↗

arXiv2026

Beating Quadratic Time--Message Trade-off in Distributed Minimum Spanning Tree Construction

We present a new distributed algorithm for computing a minimum spanning tree (MST) in the \textsf{CONGEST-KT$_{1}$} model, where messages are limited to $O(\log n)$ bits and each vertex initially knows the identifiers of its neighbors. Our algorithm exposes a two-parameter time--message trade-off: for any $0 \leq λ\leq κ\leq 1/2$, it runs in $\tilde{O}(n^λD_G + n^{1 - κ- λ} + n^{1 - 2κ+ λ} + n^{1/2})$ rounds and uses $\tilde{O}(\min\{m, n^{1 + κ}\})$ messages, where $n$, $m$, and $D_G$ are the number of vertices, edges, and thenetwork diameter, respectively. In particular, setting $(κ, λ) = (1/3, 1/6)$ yields an MST algorithm running in $\tilde{O}(n^{1/2} + n^{1/6}D_G)$ rounds with only $\tilde{O}(n^{4/3})$ messages. Under the mild assumption $D_G = O(n^{1/3})$, this is round-optimal while improving the best known message bound of $\tilde{O}(n^{3/2})$. More broadly, our algorithm breaks the quadratic time--message trade-off barrier $\mathrm{\# rounds} \cdot \mathrm{\# messages} = \tildeΩ(n^2)$, which no previous MST algorithm in the \textsf{CONGEST-KT$_{1}$} model has been able to overcome, and it does so for almost the entire range of the diameter $D_G$. As a byproduct, we also obtain new low-message broadcast, spanning-tree, and leader-election algorithms.

cs.DS↗

arXiv2026

Finding Shortest Reconfiguration Sequences on Independent Set Polytopes

We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the $1$-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in $\mathbb{R}^n$ described by $O(n)$ linear inequalities is hard to approximate within a factor of $(1-\varepsilon)\ln n$ for any constant $ε>0$, unless $P=NP$. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

cs.DS↗

arXiv2026

Finite Pinwheel Covering

In perpetual scheduling theory, the Pinwheel Covering problem asks, given $n$ frequencies $f_i$, whether there exists an infinite schedule such that every $f_i$ consecutive entries contain at most one occurrence of $i\in [n]$. This models $n$ agents taking turns at executing a job, with a recovery period before working again. Pinwheel Covering is, in a sense, the dual of Pinwheel Packing (also known as Pinwheel Scheduling), which similarly asks for at least one occurrence of $i$ in every $f_i$ consecutive entries. The complexity of both problems is a major open question: both are known to be in PSPACE, but PSPACE-hardness remains unknown. Recently, a finite version of Pinwheel Packing requiring only $k$ occurrences of $i\in [n]$ was introduced by [Kanellopoulos et al., SODA 2026] and proven to be strongly NP-complete. In this work we introduce $k$-Visits Covering, the analogous finite version of Pinwheel Covering, establishing strong NP-completeness even for $k=2$. As a corollary, we obtain that a generalization of Pinwheel Covering with varying frequencies is strongly NP-hard. To the best of our knowledge, this is the first strong NP-hardness result in the covering setting. We complement these results with a linear-time algorithm for $2$-Visits Covering with two distinct frequencies and a randomized polynomial-time algorithm when the number of distinct frequencies is constant. Lastly, we study the density thresholds of $k$-Visits Covering and prove that no non-trivial density bounds exist, contrasting the finite packing version.

cs.DS↗

arXiv2026

Lower Bounds for Linear Hashing via Arithmetic Kakeya

Affine modular linear hashing is one of the simplest classical hash families. For a prime $p > u$, the hash function is obtained by choosing $s,t$ uniformly from $\mathbb{Z}_p$ and mapping each key $x \in \{0,\ldots,u-1\}$ to one of $n$ bins by $h(x) = [(sx+t) \bmod p] \bmod n$. Despite its simplicity, the maximum load of linear hashing remains poorly understood. For $n$ keys hashed into $n$ bins, the best known upper bound is $O((n \log n)^{1/3})$, whereas the best known lower bound is only $Ω(\log n / \log\log n)$. We prove a lower bound of $\exp(Ω(\log n / \log\log n))$ for universes of size $n^{1+o(1)}$. Surprisingly, there is a key set for which this load holds not just in expectation, but for every random seed. The proof is driven by two simple reductions: one transfers lower bounds from a real version of linear hashing to modular linear hashing, and the other transfers arithmetic Kakeya constructions to real hashing. We further show that, for sufficiently large $p$, the expected maximum loads in the modular and real settings are essentially the same, giving an alternative route to an $n^{1/3+o(1)}$ upper bound. Finally, we show that any uniform subpolynomial upper bound for either setting would imply a polynomial-length arithmetic Kakeya conjecture and hence the Kakeya conjecture for upper Minkowski dimension.

cs.DS↗

arXiv2026

On the Instance Optimality of Bidirectional Dijkstra's Algorithm

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.

cs.DS↗

arXiv2026

Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid

The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $Ω(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $Ω(k)$ and $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.

quant-ph↗

arXiv2026

On the Parameterized Complexity of $s$-Club Cluster Edge Deletion

We study the parameterized and kernelization complexity of the \emph{\textsc{$s$-Club Cluster Edge Deletion}} problem, a distance-bounded generalization of \emph{\textsc{Cluster Edge Deletion}}. Given a graph $G=(V,E)$ and integers $k,s$, the goal is to delete at most $k$ edges so that every resulting connected component has diameter at most $s$. On the structural side, we settle an open question of Montecchiani, Ortali, Piselli, and Tappini (\emph{Theoretical Computer Science}, 2023) by proving W[1]-hardness parameterized by pathwidth plus the maximum number of allowed $s$-clubs, and consequently by treewidth plus this parameter. Thus, the diameter bound $s$ is inecessary for tractability under these parameters. In contrast, we show that dependence on $s$ is unnecessary for several structural parameters: the problem is fixed-parameter tractable when parameterized by treedepth, neighborhood diversity, or cluster vertex deletion number, generalizing known results for $s=1.$ We further prove that no polynomial kernel exists when parameterized by vertex cover, even for $s=2$. On the positive side, we present an FPT bicriteria approximation scheme for graphs excluding long induced cycles, running in time $f(k,1/ε)\cdot n^{\mathcal{O}(1)}$ and producing a solution of size at most $k$ whose components have diameter at most $(1+ε)s$. Finally, we initiate the study of the directed variant, \textsc{$s$-Club Cluster Arc Deletion}, and prove that it is W[1]-hard parameterized by $k$, even on directed acyclic graphs.

cs.DM↗