arXiv subjects

Yuni Iwamasa

Publications and source records attributed to Yuni Iwamasa.

At least 19 recordsLinked to original sources

arXiv2026

Towards discrete convex analysis over classical root systems

Discrete Convex Analysis (DCA) is a discrete analog of continuous convex analysis, originally proposed as a unified theoretical framework for efficiently solvable combinatorial optimization problems. Recently, DCA has proven to be a powerful tool across diverse fields, ranging from operations research to economics and pure mathematics. Motivated by the broad applicability of DCA, this paper establishes a unified theory of discrete convex analysis over discrete structures arising from classical root systems, extending the usual setting of the integer lattice, which essentially corresponds to type A. We adopt the vertex set of the Euclidean Coxeter complex as the primal discrete domain for L-convexity, and the root lattice as the dual discrete domain for M-convexity. Using the associated polyhedral structures, we formulate L- and M-convex functions together with notions of integrality determined by the root system. We show that local optimality guarantees global optimality for these functions. Furthermore, we establish that integral L-convex functions and integral M-convex functions correspond one-to-one via the discrete Fenchel--Legendre conjugate, thereby extending the conjugacy in the original DCA from type A to all classical root systems.

math.CO↗

arXiv2025

Homotopy types of Hom complexes of graph homomorphisms whose codomains are cycles

For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom complexes play an important role in an algebro-topological approach to the graph coloring problem. It is known that $\mathrm{Hom}(G,H)$ is homotopy equivalent to a disjoint union of points and circles when both $G$ and $H$ are cycles. We generalize this known result by showing that the same holds whenever $G$ is connected and $H$ is a cycle. To this end, we explicitly construct the universal cover of each connected component of $\mathrm{Hom}(G,H)$ and prove that it is contractible. Additionally, we provide a simple criterion to determine whether the connected component containing a given homomorphism is homotopy equivalent to a point or circle.

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arXiv2025

Algorithmic aspects of semistability of quiver representations

We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding the semistability and $σ$-semistability, finding the maximizers of King's criterion, and computing the Harder--Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King's criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the $σ$-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.

math.OC↗

arXiv2025

A general framework for finding diverse solutions via network flow and its applications

In this paper, we present a general framework for efficiently computing diverse solutions to combinatorial optimization problems. Given a problem instance, the goal is to find $k$ solutions that maximize a specified diversity measure; the sum of pairwise Hamming distances or the size of the union of the $k$ solutions. Our framework applies to problems satisfying two structural properties: (i) All solutions are of equal size and (ii) the family of all solutions can be represented by a surjection from the family of ideals of some finite poset. Under these conditions, we show that the problem of computing $k$ diverse solutions can be reduced to the minimum cost flow problem and the maximum $s$-$t$ flow problem. As applications, we demonstrate that both the unweighted minimum $s$-$t$ cut problem and the stable matching problem satisfy the requirements of our framework. By utilizing the recent advances in network flows algorithms, we improve the previously known time complexities of the diverse problems, which were based on submodular function minimization.

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arXiv2024

Computing diverse pair of solutions for tractable SAT

In many decision-making processes, one may prefer multiple solutions to a single solution, which allows us to choose an appropriate solution from the set of promising solutions that are found by algorithms. Given this, finding a set of \emph{diverse} solutions plays an indispensable role in enhancing human decision-making. In this paper, we investigate the problem of finding diverse solutions of Satisfiability from the perspective of parameterized complexity with a particular focus on \emph{tractable} Boolean formulas. We present several parameterized tractable and intractable results for finding a diverse pair of satisfying assignments of a Boolean formula. In particular, we design an FPT algorithm for finding an ``almost disjoint'' pair of satisfying assignments of a $2$CNF formula.

cs.DS↗

arXiv2024

Basis sequence reconfiguration in the union of matroids

Given a graph $G$ and two spanning trees $T$ and $T'$ in $G$, Spanning Tree Reconfiguration asks whether there is a step-by-step transformation from $T$ to $T'$ such that all intermediates are also spanning trees of $G$, by exchanging an edge in $T$ with an edge outside $T$ at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of $T$ and $T'$. Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of $c \log n$ for some constant $c > 0$, where $n$ is the total size of the ground sets of the input matroids.

math.CO↗

arXiv2024

On the sizes of BDDs and ZDDs representing matroids

Matroids are often represented as oracles since there are no unified and compact representations for general matroids. This paper initiates the study of binary decision diagrams (BDDs) and zero-suppressed binary decision diagrams (ZDDs) as relatively compact data structures for representing matroids in a computer. This study particularly focuses on the sizes of BDDs and ZDDs representing matroids. First, we compare the sizes of different variations of BDDs and ZDDs for a matroid. These comparisons involve concise transformations between specific decision diagrams. Second, we provide upper bounds on the size of BDDs and ZDDs for several classes of matroids. These bounds are closely related to the number of minors of the matroid and depend only on the connectivity function or pathwidth of the matroid, which deeply relates to the classes of matroids called strongly pigeonhole classes. In essence, these results indicate upper bounds on the number of minors for specific classes of matroids and new strongly pigeonhole classes.

math.CO↗

arXiv2023

Finding a Maximum Restricted $t$-Matching via Boolean Edge-CSP

The problem of finding a maximum $2$-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum $t$-matching which excludes specified complete $t$-partite subgraphs, where $t$ is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete $t$-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most $2t-1$. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum $2$-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of Bérczi and Végh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most $t+1$. Our algorithms are obtained from exploiting the discrete structure of restricted $t$-matchings and employing an algorithm for the Boolean edge-CSP.

math.CO↗

arXiv2023

Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

We address the computation of the degrees of minors of a noncommutative symbolic matrix of form \[ A[c] := \sum_{k=1}^m A_k t^{c_k} x_k, \] where $A_k$ are matrices over a field $\mathbb{K}$, $x_i$ are noncommutative variables, $c_k$ are integer weights, and $t$ is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of $A[c]$ of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes.

math.CO↗

arXiv2023

Reconfiguration of Time-Respecting Arborescences

An arborescence, which is a directed analogue of a spanning tree in an undirected graph, is one of the most fundamental combinatorial objects in a digraph. In this paper, we study arborescences in digraphs from the viewpoint of combinatorial reconfiguration, which is the field where we study reachability between two configurations of some combinatorial objects via some specified operations. Especially, we consider reconfiguration problems for time-respecting arborescences, which were introduced by Kempe, Kleinberg, and Kumar. We first prove that if the roots of the initial and target time-respecting arborescences are the same, then the target arborescence is always reachable from the initial one and we can find a shortest reconfiguration sequence in polynomial time. Furthermore, we show if the roots are not the same, then the target arborescence may not be reachable from the initial one. On the other hand, we show that we can determine whether the target arborescence is reachable form the initial one in polynomial time. Finally, we prove that it is NP-hard to find a shortest reconfiguration sequence in the case where the roots are not the same. Our results show an interesting contrast to the previous results for (ordinary) arborescences reconfiguration problems.

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arXiv2023

Reconfiguring (non-spanning) arborescences

In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of reconfiguring arborescences in a digraph, where an arborescence is a directed graph such that its underlying undirected graph forms a tree and all vertices have in-degree at most 1. Given two arborescences in a digraph, the goal of the problem is to determine whether there is a (reconfiguration) sequence of arborescences between the given arborescences such that each arborescence in the sequence can be obtained from the previous one by removing an arc and then adding another arc. We show that this problem can be solved in polynomial time, whereas the problem is PSPACE-complete when we restrict arborescences in a reconfiguration sequence to directed paths or relax to directed acyclic graphs. We also show that there is a polynomial-time algorithm for finding a shortest reconfiguration sequence between two spanning arborescences.

cs.DS↗

arXiv2023

Characterizations of the set of integer points in an integral bisubmodular polyhedron

In this note, we provide two characterizations of the set of integer points in an integral bisubmodular polyhedron. Our characterizations do not require the assumption that a given set satisfies the hole-freeness, i.e., the set of integer points in its convex hull coincides with the original set. One is a natural multiset generalization of the exchange axiom of a delta-matroid, and the other comes from the notion of the tangent cone of an integral bisubmodular polyhedron.

math.CO↗

arXiv2022

Quantaloidal Approach to Constraint Satisfaction

The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and polymorphisms, can be formulated abstractly inside the 2-category PFinSet of finite sets and sets of functions between them. The 2-category PFinSet is a quantaloid, and the formulation relies mainly on structure available in any quantaloid. This observation suggests a formal development of generalisations of the CSP and concomitant notions of polymorphism in a large class of quantaloids. We extract a class of optimisation problems as a special case, and show that their computational complexity can be classified by the associated notion of polymorphism.

math.CT↗

arXiv2022

Reconfiguration of colorings in triangulations of the sphere

In 1973, Fisk proved that any $4$-coloring of a $3$-colorable triangulation of the $2$-sphere can be obtained from any $3$-coloring by a sequence of Kempe-changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe-change, there exists a $4$-coloring that cannot be obtained from any $3$-coloring. In this paper, we present a characterization of a $4$-coloring of a $3$-colorable triangulation of the $2$-sphere that can be obtained from a $3$-coloring by a sequence of recoloring operations at single vertices, and a criterion for a $3$-colorable triangulation of the $2$-sphere that all $4$-colorings can be obtained from a $3$-coloring by such a sequence. Moreover, our first result can be generalized to a high-dimensional case, in which ``$4$-coloring,'' ``$3$-colorable,'' and ``$2$-sphere'' above are replaced with ``$k$-coloring,'' ``$(k-1)$-colorable,'' and ``$(k-2)$-sphere'' for $k \geq 4$, respectively. In addition, we show that the problem of deciding whether, for given two $(k+1)$-colorings, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed $k \geq 4$. Our results above can be rephrased as new results on the computational problems named {\sc $k$-Recoloring} and {\sc Connectedness of $k$-Coloring Reconfiguration Graph}, which are fundamental problems in the field of combinatorial reconfiguration.

math.CO↗

arXiv2022

Rerouting Planar Curves and Disjoint Paths

In this paper, we consider a transformation of $k$ disjoint paths in a graph. For a graph and a pair of $k$ disjoint paths $\mathcal{P}$ and $\mathcal{Q}$ connecting the same set of terminal pairs, we aim to determine whether $\mathcal{P}$ can be transformed to $\mathcal{Q}$ by repeatedly replacing one path with another path so that the intermediates are also $k$ disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when $k=2$. On the other hand, we prove that, when the graph is embedded on a plane and all paths in $\mathcal{P}$ and $\mathcal{Q}$ connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint $s$-$t$ paths as a variant. We show that the disjoint $s$-$t$ paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.

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arXiv2022

Reforming an Envy-Free Matching

We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

cs.GT↗

arXiv2022

Independent set reconfiguration on directed graphs

\textsc{Directed Token Sliding} asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. \textsc{Directed Token Sliding} is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions. In this paper, we initiate the algorithmic study of \textsc{Directed Token Sliding}. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a linear-time algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees~[Demaine et al.~TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which admits an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

cs.DS↗

arXiv2022

Reconstructing phylogenetic trees from multipartite quartet systems

A phylogenetic tree is a graphical representation of an evolutionary history of taxa in which the leaves correspond to the taxa and the non-leaves correspond to speciations. One of important problems in phylogenetic analysis is to assemble a global phylogenetic tree from small phylogenetic trees, particularly, quartet trees. {\sc Quartet Compatibility} is the problem of deciding whether there is a phylogenetic tree inducing a given collection of quartet trees, and to construct such a phylogenetic tree if it exists. It is known that {\sc Quartet Compatibility} is NP-hard and that there are only a few results known for polynomial-time solvable subclasses. In this paper, we introduce two novel classes of quartet systems, called complete multipartite quartet system and full multipartite quartet system, and present polynomial-time algorithms for {\sc Quartet Compatibility} for these systems.

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