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cs.DS: explore 187 source-linked works published from 2022 to 2026, with original documents and citations.
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Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.
In this paper, we study several statistical problems on similarity graphs in the node-arrival streaming model, including degree moments, diversity index, degree-moment sampling, and diversity sampling. We develop constant-pass, sublinear-space streaming algorithms for these problems and establish space lower bounds that nearly match the upper bounds in their dependence on the stream length.
Karger's elegant random contraction algorithm for finding a global mincut in a graph has been highly influential. More recent work has obtained several different (nonuniform) random contraction algorithms for mincut in hypergraphs and hedgegraphs. Motivated by the conceptual goal of understanding these algorithms in a unified fashion, we study random contraction algorithms for finding a minimum quotient of a polymatroid. We introduce the notion of quotient-bounded polymatroids and show that several existing results can be derived and understood under a common algorithmic framework for quotient-bounded polymatroids.
In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable partition always does and provides a succinct certificate for the unsolvability of an instance. Furthermore, apart from being a useful structural tool to study the problem, every stable partition corresponds to a stable half-matching, which has applications, for example, in sports scheduling and time-sharing. We establish new structural results for stable partitions and show how to enumerate all stable partitions and the cycles included in such structures efficiently. We also adapt optimality criteria from stable matchings to stable partitions and give complexity and approximability results for the problems of computing such "fair" and "optimal" stable partitions. Through this research, we contribute to a deeper understanding of stable partitions from a combinatorial point of view, as well as the computational complexity of computing "fair" or "optimal" stable half-matchings in practice, closing the gap between integral and fractional stable matchings and paving the way for further applications of stable partitions to unsolvable instances and computationally hard stable matching problems.
Two words $x,y$ of the same length are said to be \emph{parameterized equivalent} if there exists a character bijection that transforms $x$ into $y$. A word $w$ is called a parameterized square if $w$ is a concatenation of two parameterized equivalent words. Kociumaka et al. [TCS 2016] showed that in a word of length $n$ that contains $σ$ distinct characters, the number of \emph{parameterized squares} that are non-equivalent with respect to parameterized equivalence is at most $2 σ! n$. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than $σn$, which significantly improves the best-known upper bound by Kociumaka et al. Moreover, we construct a family of words containing $Ω(σn)$ non-equivalent parameterized squares, which demonstrates that the upper bound is asymptotically tight.
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.
We study the Stable Fixtures problem, a many-to-many generalisation of the classical non-bipartite Stable Roommates matching problem. Building on the foundational work of Tan on stable partitions, we extend his results to this significantly more general setting and develop a rich framework for understanding stable structures. Our main contribution, the notion of a generalised stable partition (GSP), not only characterises the solution space but also serves as a versatile tool for ordinal preference systems with capacity constraints. We show that a GSP can be computed efficiently and can provide an elegant representation of key aspects of a preference system. Leveraging a connection to stable half-matchings, we also establish an analogous Rural Hospitals Theorem for stable half-matchings and GSPs, and connect our results to recent work on near-feasible matchings, providing a simpler algorithm and tighter analysis. Our work also addresses the computational challenges of finding optimal stable half-matchings and GSPs, presenting a flexible integer linear programming model for various objectives. Beyond theoretical insights, we conduct the first empirical analysis of random Stable Fixtures instances. Our work unifies and extends classical and recent perspectives on stability in non-bipartite stable matching and establishes new tools and techniques for stable matchings and their applications.
Subgraph isomorphism is a fundamental problem in graph analysis that seeks to find all instances of a pattern graph within a larger data graph while preserving structural relationships. This NP-complete problem is central to domains such as biological network analysis, social network mining, and quantum circuit optimization. Traditional approaches rely on backtracking algorithms like VF2, which suffer from sequential bottlenecks that limit their ability to exploit modern parallel hardware. In this work, we introduce $Δ$-Motif, a GPU-accelerated subgraph isomorphism algorithm that reformulates the task through the lens of database operations. Our key insight is to represent both data and pattern graphs in tabular form, turning subgraph isomorphism into database primitives including joins, sorts, merges, and filters. $Δ$-Motif decomposes graphs into small building blocks called motifs and systematically combines them using scalable relational operations. By leveraging mature, optimized libraries from the NVIDIA RAPIDS ecosystem and Pandas framework, our solution achieves massive parallelism while remaining portable across systems supporting standard relational primitives. Benchmarks show that $Δ$-Motif outperforms established algorithms like VF2, achieving speedups of up to $595\times$ on GPUs. We further demonstrate its impact by applying it to quantum circuit compilation, addressing a critical bottleneck in quantum computing and enabling scaling to near- and medium-term devices. Our approach democratizes high-performance graph processing by exposing it through familiar database abstractions, eliminating the need for low-level programming while delivering exceptional computational efficiency.
Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either unachievable or comes at the cost of leaving agents unmatched. In these cases, algorithmicists and market designers face a critical question: how should instability be measured and distributed among participants? Existing approaches to "almost-stable" matchings focus on aggregate measures, minimising the number of blocking pairs or the count of agents involved in blocking pairs. However, these objectives can result in concentrated instability on a few agents, raising concerns about fairness and incentives to deviate. We introduce a fairness-oriented approach to approximate stability based on the minimax principle: we seek matchings that minimise the maximum number of blocking pairs any agent is in. Equivalently, we minimise the maximum number of agents that anyone has justified envy towards. This distributional objective protects the worst-off agents from bearing a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees with respect to the distribution of instability prove computationally intractable: we show that it is NP-complete to decide whether a matching exists in which no agent is in more than one blocking pair, even when preference lists are bounded. This result applies to both Stable Roommates and maximum cardinality Stable Marriage. On the positive side, we provide polynomial-time algorithms when agents rank at most two others, and present approximation algorithms and integer programs for general settings. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees on justified envy and computational feasibility in matching market design.
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.
This paper presents a simple framework that settles the complexity of Multi-Agent Path Finding (MAPF) on trees across standard objectives - distance, makespan, and flowtime - for both labeled and colored variants. In MAPF, agents occupy the vertices of a graph and must move to target vertices without collisions while optimizing a given objective. In the labeled case, the agents are distinct and have respective targets; in the colored case, agents of the same color are interchangeable. While many MAPF variants are known to be intractable, several basic cases on trees have remained open. We prove NP-hardness on trees for both labeled and 2-colored MAPF under all three objectives. In particular, we resolve the classical Pebble Motion problem, where one pebble moves at a time to an adjacent empty vertex and the goal is to minimize the total number of moves. Despite being one of the most basic discrete motion models, its complexity on trees had remained open for several decades. Moreover, for colored Pebble Motion, we give the first hardness result on any graph class, already with two colors, which is tight. All of these results are established through the hardness of Stack Rearrangement, itself posed as an open problem, which asks to optimally rearrange items stored in stacks, and which we also prove to be NP-hard. Notably, the connection to stacks yields hardness already on very simple trees - subdivided stars - across all problems. Together, these results reveal a common tractability barrier that permeates several fundamental motion models, thereby unifying and strengthening prior hardness results.
The suffix array ($\SA$) and inverse suffix array ($\ISA$) are fundamental data structures in string algorithms. For a text $T[0 \dd n)$ over an alphabet $[0 \dd σ)$, explicitly storing either structure requires $Θ(n\log n)$ bits, whereas the text itself requires only $n\logσ$ bits. Classical compressed indexes, including the FM-index of Ferragina and Manzini [FOCS 2000] and the compressed suffix array of Grossi and Vitter [STOC 2000], reduce the space to $O(n\logσ)$ bits while supporting both $\SA$ and $\ISA$ queries efficiently. Moreover, essentially all known approaches yield nearly identical space--time trade-offs for the two operations. This raises a natural question: \emph{under the same asymptotic space bound, do $\SA$ and $\ISA$ queries have the same inherent query-time complexity?} We provide strong evidence that they do not. Achieving $\log^{o(1)} n$ query time for $\SA$ using $O(n\logσ)$ bits would require a major breakthrough in computational geometry, by a straightforward consequence of a hardness result of Chien {\it et al.} [Algorithmica 2015]. In contrast, we give a near-succinct encoding using $n\logσ+o(n\logσ)+O(n)$ bits that supports $t_{\ISA}=O!\left(τ+\frac{\log\log n}{\log\logσ}\right)$, where $τ=ω(1)$ can grow arbitrarily slowly, and $t_{\SA}=O(t_{\ISA}τ\log_σ n)$. The leading $n\logσ$ bits are used solely to store the text in packed form, enabling optimal-time substring extraction and efficient packed pattern searching. Building on Sadakane [SODA 2002], we also give a suffix-tree encoding auxiliary to the text that uses only $O(n\log\log\logσ)$ bits and supports standard operations in polylogarithmic time. For growing alphabets, this auxiliary space is asymptotically smaller than the space required to store the text itself.
We study the minimum-weight mixed dominating set problem on threshold graphs. In this problem, vertices and edges have weights, and the goal is to find a mixed set of minimum total weight that dominates every vertex and edge of the graph. We first show that arbitrary weights can be reduced to non-negative weights without changing the asymptotic running time. By adapting a reduction to the minimum-weight edge cover given in Ferrarini, Kober, Lancini, and Yuditsky, we obtain an $\mathcal{O}(n^5)$-time algorithm for the minimum weight mixed dominating set problem on threshold graphs.
Daviaud, Guillon, and Merlet proved that comparison of max-plus automata is undecidable under a fixed state bound of 553 and explicitly left the range from 2 to 552 states open. We resolve the two-state endpoint. More strongly, given an arbitrary finite max-plus automaton $A$ and a max-plus automaton $B$ with at most two states, it is decidable whether $[\![A]\!](w)\leq [\![B]\!](w)$ for every word $w$. The structural reason is a one-dimensional projective normal form for two-state dynamics. Outside an effective bounded region, a transition has one of three tail behaviors: it propagates the unbounded projective gap with gap-independent height increment, forgets the gap with gap-independent height increment, or reads the gap magnitude into the height increment and then forgets it. In particular, any transition whose output depends on the unbounded gap necessarily destroys that gap. This yields an exact one-counter realization of $B$. Effective semilinearity of context-free Parikh images then reduces comparison to Presburger arithmetic. As a consequence, two-state max-plus comparison, equivalence, and positivity are decidable.
For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.
Let $G\in\mathbb{R}^{M\times N}$ have independent standard Gaussian entries. For a fixed margin $κ\in\mathbb{R}$, the asymmetric binary perceptron asks for $σ\in\{\pm1\}^N$ such that $Gσ/\sqrt{N}\geκ\mathbf{1}_M$. We study the online version of this problem, in which the columns of $G$ arrive sequentially and each sign must be chosen irrevocably before future columns are revealed. We determine the exact threshold $α_{\mathrm{on}}(κ)$ for every fixed $κ$: for $M/N\toα$ with $α<α_{\mathrm{on}}(κ)$, there is a deterministic online algorithm, using $O(MN)$ arithmetic operations and polynomial bit complexity, that succeeds with high probability, while for $α>α_{\mathrm{on}}(κ)$, no online algorithm succeeds with high probability. The threshold is characterized by a one-dimensional stochastic control problem for Brownian motion. The main difficulty is to upgrade a single-coordinate Brownian limit to simultaneous feasibility of all $M=Θ(N)$ constraints, which we do with half-line monotonicity and a short final correction block. At zero margin, we give a computer-assisted proof that $0.32747<α_{\mathrm{on}}(0)<0.36664$. In particular, every density below $0.32747$ is achievable online by such an algorithm, more than tripling the best density previously proved attainable by any polynomial-time algorithm, online or offline (the previous bound was $α\le0.1$, due to Li, Schramm, and Zhou). As $κ\to+\infty$, the online threshold agrees to first order with the offline storage capacity. As $κ\to-\infty$, it has the same asymptotic scale as the best known offline polynomial-time guarantee, while the storage capacity is larger by a factor of order $κ^2$.
We study the low-degree Steiner forest decomposition. Given a graph $G=(V,E)$ and a terminal set $U\subseteq V$, the standard decomposition returns a set $X\subseteq V$ of size at most $|U|/2$ and a forest $T\subseteq G-X$ of maximum degree $Δ$ such that, for every connected component $C$ of $G-X$, some connected component of $T$ contains all terminals in $U\cap V(C)$. This is the central decomposition behind several connectivity oracles under vertex failures [DP20, LS22, LW24]. The state-of-the-art algorithms either take $O(mn\log n)$ time with degree bound $4$ [DP20], or take $m^{1+o(1)}$ time with the weaker degree bound $O(\log^{2}n)$ [LW24]. We show that if $T$ is allowed to contain vertices of $X$, then a degree-$4$ decomposition can be computed by a very simple algorithm in $O(mα(n))$ time. Further, we show that this relaxed decomposition is equally useful for constructing connectivity oracles under vertex failures. As a consequence, we obtain a deterministic connectivity oracle under $d$ vertex failures with $\tilde{O}(m)$ space, $\tilde{O}(md_\star)$ preprocessing time ($d_\star$ is an upper bound on the number of failed vertices), $\tilde{O}(d^{2})$ update time, and $O(d)$ query time. Up to polylogarithmic factors, this oracle strictly improves all known oracles; in particular, it removes the $n^{o(1)}$ factors from the preprocessing and update times of [LS22, LW24].
Navigability is a fundamental property of graph-based search structures and plays an important role in the analysis of nearest-neighbor algorithms. Informally, a graph is navigable if, from any current point and toward any desired target, there is always an outgoing edge that moves strictly closer to that target. While this property provides a strong guarantee for greedy search, it can be inherently expensive: in the worst case, navigable graphs require $Ω(n^{3/2})$ edges, where $n$ is the size of the dataset. Recently, Avi and Musco introduced $(1-ε)$-almost navigability, a natural relaxation in which, from every current point, such a progress-making edge is required for almost all targets, while an $ε$ fraction of targets may fail this condition \cite{avimusco2026almost}. They showed that every dataset admits such a graph with $O(n/ε)$ edges. In this work, we prove a matching lower bound, establishing the optimality of their construction and giving a complete characterization of the sparsity achievable by almost-navigable graphs. For most values of $ε$, our hard instances lie in Euclidean spaces of polylogarithmic dimension, across the full worst-case range of $ε$, dimension $d=O(\sqrt{n}\log^{3/2} n)$ suffices. The same construction also sharpens our understanding of ordinary navigability: in the latter dimension, we exhibit datasets for which every navigable graph has $Ω(n^{3/2})$ edges. Our proof reveals a surprising connection between almost navigability and the classical Zarankiewicz problem of constructing dense graphs with limited pairwise neighborhood overlap. This connection lets us translate extremal graph constructions into hard geometric instances for navigation, linking two seemingly different notions of graph sparsity.
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 $2^{n/2+o(n)}$.