arXiv subjects

Yutaro Yamaguchi

Publications and source records attributed to Yutaro Yamaguchi.

At least 19 recordsLinked to original sources

arXiv2026

Exact Matching in Matrix Multiplication Time

Let $A_0,A_1\in\mathbf{F}^{n\times n}$ be square matrices over a finite field $\mathbf{F}$ and consider the matrix pencil $A_0+yA_1$ with indeterminate $y$. We observe that, once $A_0+λA_1$ is nonsingular for some $λ\in\mathbf{F}$, the polynomial $\det(A_0+yA_1)$ can be reconstructed by computing one determinant, one inverse matrix, and the characteristic polynomial of a single matrix. Consequently, this determinant polynomial can be computed in $\mathrm{O}(n^ω)$ field operations, avoiding the polylogarithmic overhead of a general polynomial-matrix determinant algorithm in this special setting. Applying this observation to random evaluation of the Tutte matrix of a graph, we obtain a matrix-multiplication-time randomized algorithm for the so-called exact matching problem. Specifically, one can decide, simultaneously for all $k$, whether a given $0/1$-weighted graph has a perfect matching of weight exactly $k$ in $\mathrm{O}(n^ω)$ field operations, where $n$ denotes the number of vertices in the graph. We also discuss the analogous extension to the exact linear matroid parity problem and its consequences for a perfect packing of Mader's $\mathcal{S}$-paths of minimum total length and for a shortest cycle through three specified vertices.

cs.DS↗

arXiv2026

A Linear-Time Algorithm for Finding an Odd/Even Cycle Through Two Specified Vertices

We present a deterministic linear-time algorithm for finding an odd/even cycle through two specified vertices in an undirected graph. This is shown in a generalized form as follows: Let $Γ$ be any group in which every element is of order at most $2$. For a given $Γ$-labeled graph with two specified vertices (or edges), we can determine in linear time whether there exist two cycles with distinct labels that are through both of the two specified vertices (or edges), and find such cycles if yes.

cs.DS↗

arXiv2026

Approximate Maintenance of Maximum Subarray Sum in the Sliding Window Model

In the sliding window model, we are required to maintain the target statistics over the most recent $n$ elements of a data stream, which is captured by a window of size $n$ sliding over the data stream. Exact computation usually requires space linear in $n$, and the central goal is approximate maintenance using sublinear space. In this paper, we study the problem of maintaining the maximum subarray sum in the sliding window model. While the classical Kadane's algorithm computes the exact answer using constant space in the static setting, it does not extend directly, because a new element makes the oldest one expire, which may invalidate the optimal subarray so far. Our first observation is that the so-called Smooth Histogram framework can lead to a constant-factor approximation (in the sense of relative error) using $O((\log n)^2)$ bits of space. We then refine this framework accordingly, which enables for any $ε> 0$ to maintain a $(1 \pm ε)$-approximation using $O(ε^{-1}(\log n)^2)$ bits of space and $O(ε^{-1}\log n)$ operations per update. The space complexity is asymptotically optimal.

cs.DS↗

arXiv2026

A hierarchy of edge-weight symmetries in perfect matchings

Motivated by the exact weight perfect matching problem and recent parameterized algorithms for finding an $\ell$-th smallest perfect matching, we study structural properties of edge-weight symmetries in graphs. Recent work by El Maalouly et al. (ESA 2025) showed that excluding all perfect matchings whose weight is at most the $(\ell - 1)$-th smallest possible value in the graph requires fixing at most $2(\ell-1)$ edges in non-bipartite graphs and at most $\ell-1$ edges in bipartite graphs. A natural open question is whether fixing a single edge is always sufficient to shift the extreme (minimum or maximum) weight of a perfect matching when the global minimum and maximum weights differ. To address this, we define and analyze a hierarchy of progressively weaker edge-weight properties: node-induced weights, even walk and cycle symmetries, perfect matching equality, and the edge min-max property. We derive a basic hierarchy among these conditions and show that they become equivalent in bipartite graphs. For general graphs, we provide tight structural characterizations, based on block and tight cut decompositions, under which even cycle symmetry and perfect matching equality force node-induced weights. Finally, we resolve the motivating open question in the negative by constructing a matching-covered non-bipartite graph that satisfies the edge min-max property (every edge is contained in a minimum-weight perfect matching and a maximum-weight one) but violates perfect matching equality (all perfect matchings have the same weight). This counterexample shows that a single edge is not always sufficient to eliminate all minimum-weight or maximum-weight perfect matchings, thereby proving the tightness of the $2(\ell-1)$ bound for $\ell=2$. We also discuss extensions of this framework to $b$-factors and arborescences.

math.CO↗

arXiv2026

Regret Analysis of Sleeping Competing Bandits

The Competing Bandits framework is a recently emerging area that integrates multi-armed bandits in online learning with stable matching in game theory. While conventional models assume that all players and arms are constantly available, in real-world problems, their availability can vary arbitrarily over time. In this paper, we formulate this setting as Sleeping Competing Bandits. To analyze this problem, we naturally extend the regret definition used in existing competing bandits and derive regret bounds for the proposed model. We propose an algorithm that simultaneously achieves an asymptotic regret bound of $\mathrm{O}\left(NK\log T_{i}/Δ^2\right)$ under reasonable assumptions, where $N$ is the number of players, $K$ is the number of arms, $T_{i}$ is the number of rounds of each player $p_i$, and $Δ$ is the minimum reward gap. We also provide a regret lower bound of $\mathrmΩ\left( N(K-N+1)\log T_{i}/Δ^2 \right)$ under the same assumptions. This implies that our algorithm is asymptotically optimal in the regime where the number of arms $K$ is relatively larger than the number of players $N$.

cs.LG↗

arXiv2026

Envy-Free School Redistricting Between Two Groups

We study an application of fair division theory to school redistricting. Procaccia, Robinson, and Tucker-Foltz (SODA 2024) recently proposed a mathematical model to generate redistricting plans that provide theoretically guaranteed fairness among demographic groups of students. They showed that an almost proportional allocation can be found by adding $O(g \log g)$ extra seats in total, where $g$ is the number of groups. In contrast, for three or more groups, adding $o(n)$ extra seats is not sufficient to obtain an almost envy-free allocation in general, where $n$ is the total number of students. In this paper, we focus on the case of two groups. We introduce a relevant relaxation of envy-freeness, termed 1-relaxed envy-freeness, which limits the capacity violation not in total but at each school to at most one. We show that there always exists a 1-relaxed envy-free allocation, which can be found in polynomial time.

cs.GT↗

arXiv2026

LZBE: an LZ-style compressor supporting $O(\log n)$-time random access

An LZ-like factorization of a string divides it into factors, each being either a single character or a copy of a preceding substring. While grammar-based compression schemes support efficient random access with space linear in the compressed size, no comparable guarantees are known for general LZ-like factorizations. This limitation motivated restricted variants such as LZ-End [Kreft and Navarro, 2013] and height-bounded LZ (LZHB) [Bannai et al., 2024], which trade off some compression efficiency for faster access. In this paper, we introduce LZ-Begin-End (LZBE), a new LZ-like variant in which every copy factor must refer to a contiguous sequence of preceding factors. This structural restriction ensures that any context-free grammar can be transformed into an LZBE factorization of the same size. We further study the greedy LZBE factorization, which selects each copy factor to be as long as possible while processing the input from left to right, and show that it can be computed in linear time. Moreover, we exhibit a family of strings for which the greedy LZBE factorization is asymptotically smaller than the smallest grammar. These results demonstrate that the LZBE scheme is strictly more expressive than grammar-based compression in the worst case. To support fast queries, we propose a data structure for LZBE-compressed strings that permits O(log n)-time random access within space linear in the compressed size, where n is the length of the input string.

cs.DS↗

arXiv2026

Computational Complexity of Swish

Swish is a card game in which players are given cards having symbols (hoops and balls), and find a valid superposition of cards, called a "swish." Dailly, Lafourcade, and Marcadet (FUN 2024) studied a generalized version of Swish and showed that the problem is solvable in polynomial time with one symbol per card, while it is NP-complete with three or more symbols per card. In this paper, we resolve the previously open case of two symbols per card, which corresponds to the original game. We show that Swish is NP-complete for this case. Specifically, we prove the NP-hardness when the allowed transformations of cards are restricted to a single (horizontal or vertical) flip or 180-degree rotation, and extend the results to the original setting allowing all three transformations. In contrast, when neither transformation is allowed, we present a polynomial-time algorithm. Combining known and our results, we establish a complete characterization of the computational complexity of Swish with respect to both the number of symbols per card and the allowed transformations.

cs.DS↗

arXiv2025

Matroid Intersection under Minimum Rank Oracle

In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the ranks of the given set in the two matroids. For the unweighted matroid intersection problem, we show how to construct a necessary part of the exchangeability graph, which enables us to emulate the standard augmenting path algorithm. For the weighted problem, the tractability is open in general. Nevertheless, we describe several special cases where tractability can be achieved, and we discuss potential approaches and the challenges encountered. On the positive side, we present a solution for the case where no circuit of one matroid is contained within a circuit of the other. Additionally, we propose a fixed-parameter tractable algorithm, parameterized by the maximum size of a circuit of one matroid. We also show that a lexicographically maximal common independent set can be found by the same approach, which leads to a nontrivial approximation ratio for finding a maximum-weight common independent set. On the negative side, we prove that the approach employed for the tractable cases above involves an NP-hard problem in the general case. We also show that if we consider the generalization to polymatroid intersection, even the unweighted problem is hard under the minimum rank oracle.

cs.DS↗

arXiv2025

Finding One Local Optimum Is Easy -- but What About Two?

The class PLS (Polynomial Local Search) captures the complexity of finding a solution that is locally optimal and has proven to be an important concept in the theory of local search. It has been shown that local search versions of various combinatorial optimization problems, such as Maximum Independent Set and Max Cut, are complete for this class. Such computational intractability typically arises in local search problems allowing arbitrary weights; in contrast, for unweighted problems, locally optimal solutions can be found in polynomial time under standard settings. In this paper, we pursue the complexity of local search problems from a different angle: We show that computing two locally optimal solutions is NP-hard for various natural unweighted local search problems, including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut. We also discuss several tractable cases for finding two (or more) local optimal solutions.

cs.DS↗

arXiv2025

Rainbow Arborescence Conjecture

The famous Ryser--Brualdi--Stein conjecture asserts that every $k \times k$ Latin square contains a partial transversal of size $k-1$. Since its appearance, the conjecture has attracted significant interest, leading to several proposed generalizations. One of the most notable of these, by Aharoni, Kotlar, and Ziv, conjectures that $k$ disjoint common bases of two matroids of rank $k$ have a common independent partial transversal of size $k-1$. Although simple counterexamples show that the size $k-1$ above cannot be improved to $k$ (i.e., a transversal instead of a partial transversal), it is remarkable that no such counterexample is known for the special case of spanning arborescences. This motivated the formulation of the Rainbow Arborescence Conjecture: any graph on $n$ vertices formed by the union of $n-1$ spanning arborescences contains an arborescence using exactly one arc from each. We prove several partial results on this conjecture. We show that the computational problem of testing the existence of such an arborescence with a fixed root is NP-complete, verify the conjecture in several special cases, and study relaxations of the problem. In particular, we establish the validity of the conjecture when the underlying undirected graph is a cycle; this also yields a new result on systems of distinct representatives for intervals on a cycle.

math.CO↗

arXiv2025

The Rainbow Arborescence Problem on Cycles

The rainbow arborescence conjecture posits that if the arcs of a directed graph with $n$ vertices are colored by $n-1$ colors such that each color class forms a spanning arborescence, then there is a spanning arborescence that contains exactly one arc of every color. We prove that the conjecture is true if the underlying undirected graph is a cycle.

math.CO↗

arXiv2025

Forgetting Alternation and Blossoms: A New Framework for Fast Matching Augmentation and Its Applications to Sequential/Distributed/Streaming Computation

Finding a maximum cardinality matching in a graph is one of the most fundamental problems. An algorithm proposed by Micali and Vazirani (1980) is well-known to solve the problem in $O(m\sqrt{n})$ time, which is still one of the fastest algorithms in general. While the MV algorithm itself is not so complicated and is indeed convincing, its correctness proof is extremely challenging, which can be seen from the history: after the first algorithm paper had appeared in 1980, Vazirani has made several attempts to give a complete proof for more than 40 years. It seems, roughly speaking, caused by the nice but highly complex structure of the shortest alternating paths in general graphs that are deeply intertwined with the so-called (nested) blossoms. In this paper, we propose a new structure theorem on the shortest alternating paths in general graphs without taking into the details of blossoms. The high-level idea is to forget the alternation (of matching and non-matching edges) as early as possible. A key ingredient is a notion of alternating base trees (ABTs) introduced by Izumi, Kitamura, and Yamaguchi (2024) to develop a nearly linear-time distributed algorithm. Our structure theorem refines the properties of ABTs exploited in their algorithm, and we also give simpler alternative proofs for them. Based on our structure theorem, we propose a new algorithm, which is slightly slower but more implementable and much easier to confirm its correctness than the MV algorithm. As applications of our framework, we also present new $(1 - ε)$-approximation algorithms in the distributed and semi-streaming settings. Both algorithms are deterministic, and substantially improve the best known upper bounds on the running time. The algorithms are built on the top of a novel framework of amplifying approximation factors of given matchings, which is of independent interest.

cs.DS↗

arXiv2025

Odd and Even Harder Problems on Cycle-Factors

For a graph (undirected, directed, or mixed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and algorithmic complexity. In this work, we study four variants of the problem of finding a cycle-factor subject to parity constraints: (1) all cycles are odd, (2) all cycles are even, (3) at least one cycle is odd, and (4) at least one cycle is even. These variants are considered in the undirected, directed, and mixed settings. We show that all but the fourth problem are NP-complete in all settings, while the complexity of the fourth one remains open for the directed and undirected cases. We also show that in mixed graphs, even deciding the existence of any cycle factor is NP-complete.

cs.DS↗

arXiv2025

A Nearly Linear-Time Distributed Algorithm for Maximum Cardinality Matching

In this paper, we propose a randomized $\tilde{O}(μ(G))$-round algorithm for the maximum cardinality matching problem in the CONGEST model, where $μ(G)$ means the maximum size of a matching of the input graph $G$. The proposed algorithm substantially improves the current best worst-case running time. The key technical ingredient is a new randomized algorithm of finding an augmenting path of length $\ell$ with high probability within $\tilde{O}(\ell)$ rounds, which positively settles an open problem left in the prior work by Ahmadi and Kuhn [DISC'20]. The idea of our augmenting path algorithm is based on a recent result by Kitamura and Izumi [IEICE Trans.'22], which efficiently identifies a sparse substructure of the input graph containing an augmenting path, following a new concept called \emph{alternating base trees}. Their algorithm, however, resorts in part to a centralized approach of collecting the entire information of the substructure into a single vertex for constructing a long augmenting path. The technical highlight of this paper is to provide a fully-decentralized counterpart of such a centralized method. To develop the algorithm, we prove several new structural properties of alternating base trees, which are of independent interest.

cs.DC↗

arXiv2024

Towards the Proximity Conjecture on Group-Labeled Matroids

Consider a matroid $M$ whose ground set is equipped with a labeling to an abelian group. A basis of $M$ is called $F$-avoiding if the sum of the labels of its elements is not in a forbidden label set $F$. Hörsch, Imolay, Mizutani, Oki, and Schwarcz (2024) conjectured that if an $F$-avoiding basis exists, then any basis can be transformed into an $F$-avoiding basis by exchanging at most $|F|$ elements. This proximity conjecture is known to hold for certain specific groups; in the case where $|F| \le 2$; or when the matroid is subsequence-interchangeably base orderable (SIBO), which is a weakening of the so-called strongly base orderable (SBO) property. In this paper, we settle the proximity conjecture for sparse paving matroids or in the case where $|F| \le 4$. Related to the latter result, we present the first known example of a non-SIBO matroid. We further address the setting of multiple group-label constraints, showing proximity results for the cases of two labelings, SIBO matroids, matroids representable over a fixed, finite field, and sparse paving matroids.

math.CO↗

arXiv2024

Approximation and FPT Algorithms for Finding DM-Irreducible Spanning Subgraphs

Finding a minimum-weight strongly connected spanning subgraph of an edge-weighted directed graph is equivalent to the weighted version of the well-known strong connectivity augmentation problem. This problem is NP-hard, and a simple $2$-approximation algorithm was proposed by Frederickson and Jájá (1981); surprisingly, it still achieves the best known approximation ratio in general. Also, Bang-Jensen and Yeo (2008) showed that the unweighted problem is FPT (fixed-parameter tractable) parameterized by the difference from a trivial upper bound of the optimal value. In this paper, we consider a generalization related to the Dulmage--Mendelsohn decompositions of bipartite graphs instead of the strong connectivity of directed graphs, and extend these approximation and FPT results to the generalized setting.

cs.DS↗

arXiv2024

Finding Spanning Trees with Perfect Matchings

We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight spanning tree among those containing a perfect matching. On the positive side, we design a simple greedy algorithm for the case when the graph is complete (or complete bipartite) and the edge weights take at most two values. On the negative side, the problem is NP-hard even when the graph is complete (or complete bipartite) and the edge weights take at most three values, or when the graph is cubic, planar, and bipartite and the edge weights take at most two values. We also consider an interesting variant. We call a tree strongly balanced if on one side of the bipartition of the vertex set with respect to the tree, all but one of the vertices have degree $2$ and the remaining one is a leaf. This property is a sufficient condition for a tree to have a perfect matching, which enjoys an additional property. When the underlying graph is bipartite, strongly balanced spanning trees can be written as matroid intersection, and this fact was recently utilized to design an approximation algorithm for some kind of connectivity augmentation problem. The natural question is its tractability in nonbipartite graphs. As a negative answer, it turns out NP-hard to test whether a given graph has a strongly balanced spanning tree or not even when the graph is subcubic and planar.

cs.DS↗