arXiv subjects

Kei Kimura

Publications and source records attributed to Kei Kimura.

At least 19 recordsLinked to original sources

arXiv2026

UTVPI-representable integer point sets: discrete convexity, polymorphisms, and pairwise closure

We study subsets of the integer lattice represented by single-variable-per-inequality (SVPI), difference-constraint (DC), unit two-variable-per-inequality (UTVPI), and two-variable-per-inequality (TVPI) systems. We relate five viewpoints: inequality representation, discrete convexity, polymorphisms, reconstruction from two-coordinate projections, and fixed points of closure operators. Our central result completely characterizes UTVPI-representability. For every set $S\subseteq\mathbb Z^n$ with $n>1$, \[ \begin{aligned} &S\text{ is UTVPI-representable}\\ &\;\Longleftrightarrow\; S\text{ is closed under the directed midpoint and median operations}\\ &\;\Longleftrightarrow\; S\text{ is integrally convex and $2$-decomposable}. \end{aligned} \] The median condition may instead be replaced by closedness under some majority operation, and the same class is the fixed-point class of a pairwise directed-midpoint closure operator. Thus, all five viewpoints yield equivalent characterizations of UTVPI-representability. In particular, $2$-decomposability is exactly the global condition needed to lift the known two-dimensional equivalence between integral convexity and UTVPI-representability to arbitrary dimension. This theorem is embedded in a broader pairwise-closure theory. For a family $F$ of operations, we define a closure operator by closing every two-coordinate projection under $F$ and joining the resulting sets. Its fixed points are precisely the sets that are both $2$-decomposable and $F$-closed, and we establish a local-to-global criterion for such characterizations. A closed-convex-hull analogue characterizes TVPI-representability. We also characterize SVPI-representability by natural multioperations, prove limitations of operation-based characterizations for several related classes, and determine the complete inclusion hierarchies in the general, Boolean, and two-dimensional settings.

cs.DM↗

arXiv2026

Towards an algebraic approach to the reconfiguration CSP

This paper investigates the reconfiguration variant of the Constraint Satisfaction Problem (CSP), referred to as the Reconfiguration CSP (RCSP). Given a CSP instance and two of its solutions, RCSP asks whether one solution can be transformed into the other via a sequence of intermediate solutions, each differing by the assignment of a single variable. RCSP has attracted growing interest in theoretical computer science, and when the variable domain is Boolean, the computational complexity of RCSP exhibits a dichotomy depending on the allowed constraint types. A notable special case is the reconfiguration of graph homomorphisms -- also known as graph recoloring -- which has been studied using topological methods. We propose a novel algebraic approach to RCSP, inspired by techniques used in classical CSP complexity analysis. Unlike traditional methods based on total operations, our framework employs partial operations to capture a reduction involving equality constraints. This perspective facilitates the extension of complexity results from Boolean domains to more general settings, demonstrating the versatility of partial operations in identifying tractable RCSP instances.

cs.DS↗

arXiv2026

MLN-EIGS: A multilayer network framework for solving Stackelberg escape interdiction games on dynamic transportation networks

Interdicting an escaping criminal with limited police resources on large-scale transportation networks is a challenging problem due to the dynamic nature of both attacker movement and defender deployment. This paper proposes \emph{MLN-EIGS}, a multilayer network-based framework for solving dynamic escape interdiction problems formulated as a Stackelberg security game. A time-expanded multilayer network is constructed to explicitly model the temporal evolution of the transportation network and the feasible movements of both the attacker and the defenders. The attacker seeks to maximize the probability of successful escape, while the defenders aim to maximize the probability of interdiction. To efficiently compute the attacker's best response, the probabilistic escape formulation is transformed into an equivalent shortest-path problem through a logarithmic transformation, enabling the use of Dijkstra's algorithm. Since the defender best-response problem is computationally intractable, an approximation defender oracle is developed to generate high-quality defender strategies on the multilayer network. The proposed MLN-EIGS framework is benchmarked against an exact mixed-integer linear programming (MILP)-based Stackelberg formulation on a large real-world transportation network. Computational experiments demonstrate that MLN-EIGS consistently achieves defender utilities that closely match those of the exact MILP approach while substantially reducing computational time. These results demonstrate that the proposed MLN-EIGS framework provides an effective, computationally efficient, and scalable alternative to exact MILP-based Stackelberg optimization for large-scale dynamic escape interdiction problems.

cs.GT↗

arXiv2026

Compatible $k$-Relaxations of Fairness and Non-Wastefulness Under Hereditary Constraints

We study two-sided matching markets under hereditary constraints, which extend beyond simple capacity limits and arise in applications such as diversity requirements and refugee resettlement. In these settings, fairness and non-wastefulness are often incompatible, and existing approaches typically address this tension by prioritizing one property at the expense of the other. We take a different approach by relaxing both properties simultaneously in a controlled and symmetric manner. We introduce two notions indexed by an integer $k$: envy-received up to $k$ peers (ER-$k$) and non-wastefulness up to $k$ objections (NW-$k$). Our main theoretical result shows that ER-$k$ and NW-$k$ are always compatible under hereditary constraints for any fixed $k$. We provide two equivalent polynomial-time algorithms to compute such matchings: a $k$-admissible cutoff algorithm and a $k$-admissible college-proposing deferred acceptance mechanism. Finally, experimental results demonstrate that even small relaxations achieve a favorable balance between fairness and non-wastefulness.

cs.GT↗

arXiv2026

Solving a Stackelberg Game on Transportation Networks in a Dynamic Crime Scenario: A Mixed Approach on Multi-Layer Networks

Interdicting a criminal with limited police resources is a challenging task as the criminal changes location over time. The size of the large transportation network further adds to the difficulty of this scenario. To tackle this issue, we consider the concept of a layered graph. At each time stamp, we create a copy of the entire transportation network to track the possible movements of both players, the attacker and the defenders. We consider a Stackelberg game in a dynamic crime scenario where the attacker changes location over time while the defenders attempt to interdict the attacker on his escape route. Given a set of defender strategies, the optimal attacker strategy is determined by applying Dijkstra's algorithm on the layered networks. Here, the attacker aims to minimize while the defenders aim to maximize the probability of interdiction. We develop an approximation algorithm on the layered networks to find near-optimal strategy for defenders. The efficacy of the developed approach is compared with the adopted MILP approach. We compare the results in terms of computational time and solution quality. The quality of the results demonstrates the need for the developed approach, as it effectively solves the complex problem within a short amount of time.

cs.AI↗

arXiv2025

A New Relaxation of Fairness in Two-Sided Matching Respecting Acquaintance Relationships

Two-sided matching, such as matching between students and schools, has been applied to various aspects of real life and has been the subject of much research, however, it has been plagued by the fact that efficiency and fairness are incompatible. In particular, Pareto efficiency and justified-envy-freeness are known to be incompatible even in the simplest one-to-one matching, i.e., the stable marriage problem. In previous research, the primary approach to improving efficiency in matchings has been to tolerate students' envy, thereby relaxing fairness constraints. In this study, we take a different approach to relaxing fairness. Specifically, it focuses on addressing only the envy that students may experience or prioritize more highly and seeks matchings without such envy. More specifically, this study assumes that envy towards students who are not acquaintances has less impact compared to envy towards students who are acquaintances. Accordingly, we assume that the students know each other or not, represented by an undirected graph, and define a local envy as a justified envy toward an acquaintance or a neighbor in the graph. We then propose the property that there is no local envy as a new relaxed concept of fairness, called local envy-freeness. We analyze whether Pareto-efficient matching can be achieved while maintaining local envy-freeness by meaningfully restricting the graph structure and the school's preferences. To analyze in detail the fairness that can achieve Pareto-efficient matching, we introduce a local version of the relaxed fairness recently proposed by Cho et al. (AAMAS 2024), which parameterizes the level of local envy-freeness by nonnegative integers. We then clarify the level of local envy-freeness that can be achieved by Pareto-efficient mechanisms for graphs that are ``close'' to trees and single-peaked preferences on the graphs.

cs.GT↗

arXiv2025

Homotopy types of Hom complexes of graph homomorphisms whose codomains are square-free

Given finite simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex having the graph homomorphisms $G\to H$ as the vertices. We determine the homotopy type of each connected component of $\mathrm{Hom}(G,H)$ when $H$ is square-free, meaning that it does not contain the $4$-cycle graph $C_4$ as a subgraph. Specifically, for a connected $G$ and a square-free $H$, we show that each connected component of $\mathrm{Hom}(G,H)$ is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism $f\colon G\to H$ to a square-free $H$, one can determine the homotopy type of the connected component of $\mathrm{Hom}(G,H)$ containing $f$ algorithmically.

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.

math.CO↗

arXiv2025

Whoever Said Money Won't Solve All Your Problems? Weighted Envy-free Allocation with Subsidy

We explore solutions for fairly allocating indivisible items among agents assigned weights representing their entitlements. Our fairness goal is weighted-envy-freeness (WEF), where each agent deems their allocated portion relative to their entitlement at least as favorable as any others relative to their own. Often, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work relied on characterizations of unweighted envy-freeness (EF), that fail in the weighted setting. This makes our new setting challenging. We present polynomial-time algorithms that compute WEF allocations with a guaranteed upper bound on total subsidy for monotone valuations and various subclasses thereof. We also present an efficient algorithm to compute a fair allocation of items and money, when the budget is not enough to make the allocation WEF. This algorithm is new even for the unweighted setting.

cs.GT↗

arXiv2024

Parameterized Voter Relevance in Facility Location Games with Tree-Shaped Invitation Graphs

Diffusion mechanism design, which investigate how to incentivise agents to invite as many colleagues to a multi-agent decision making as possible, is a new research paradigm at the intersection between microeconomics and computer science. In this paper we extend traditional facility location games into the model of diffusion mechanism design. Our objective is to completely understand to what extent of anonymity/voter-relevance we can achieve, along with strategy-proofness and Pareto efficiency when voters strategically invite collegues. We define a series of anonymity properties applicable to the diffusion mechanism design model, as well as parameterized voter-relevance properties for guaranteeing reasonably-fair decision making. We obtained two impossibility theorems and two existence theorems, which partially answer the question we have raised in the beginning of the paper

cs.GT↗

arXiv2024

Weighted Envy-free Allocation with Subsidy

We consider the problem of fair allocation of indivisible items with subsidies when agents have weighted entitlements. After highlighting several important differences from the unweighted case, we present several results concerning weighted envy-freeability including general characterizations, algorithms for achieving and testing weighted envy-freeability, lower and upper bounds of the amount of subsidies for envy-freeable allocations, and algorithms for achieving weighted envy-freeability along with other properties.

cs.GT↗

arXiv2024

Analyzing Incentives and Fairness in Ordered Weighted Average for Facility Location Games

Facility location games provide an abstract model of mechanism design. In such games, a mechanism takes a profile of $n$ single-peaked preferences over an interval as an input and determines the location of a facility on the interval. In this paper, we restrict our attention to distance-based single-peaked preferences and focus on a well-known class of parameterized mechanisms called ordered weighted average methods, which is proposed by Yager in 1988 and contains several practical implementations such as the standard average and the Olympic average. We comprehensively analyze their performance in terms of both incentives and fairness. More specifically, we provide necessary and sufficient conditions on their parameters to achieve strategy-proofness, non-obvious manipulability, individual fair share, and proportional fairness, respectively.

cs.GT↗

arXiv2024

Multi-Stage Generalized Deferred Acceptance Mechanism: Strategyproof Mechanism for Handling General Hereditary Constraints

The theory of two-sided matching has been extensively developed and applied to many real-life application domains. As the theory has been applied to increasingly diverse types of environments, researchers and practitioners have encountered various forms of distributional constraints. Arguably, the most general class of distributional constraints would be hereditary constraints; if a matching is feasible, then any matching that assigns weakly fewer students at each college is also feasible. However, under general hereditary constraints, it is shown that no strategyproof mechanism exists that simultaneously satisfies fairness and weak nonwastefulness, which is an efficiency (students' welfare) requirement weaker than nonwastefulness. We propose a new strategyproof mechanism that works for hereditary constraints called the Multi-Stage Generalized Deferred Acceptance mechanism (MS-GDA). It uses the Generalized Deferred Acceptance mechanism (GDA) as a subroutine, which works when distributional constraints belong to a well-behaved class called hereditary M$^\natural$-convex set. We show that GDA satisfies several desirable properties, most of which are also preserved in MS-GDA. We experimentally show that MS-GDA strikes a good balance between fairness and efficiency (students' welfare) compared to existing strategyproof mechanisms when distributional constraints are close to an M$^\natural$-convex set.

cs.GT↗

arXiv2024

Online $\mathrm{L}^{\natural}$-Convex Minimization

An online decision-making problem is a learning problem in which a player repeatedly makes decisions in order to minimize the long-term loss. These problems that emerge in applications often have nonlinear combinatorial objective functions, and developing algorithms for such problems has attracted considerable attention. An existing general framework for dealing with such objective functions is the online submodular minimization. However, practical problems are often out of the scope of this framework, since the domain of a submodular function is limited to a subset of the unit hypercube. To manage this limitation of the existing framework, we in this paper introduce the online $\mathrm{L}^{\natural}$-convex minimization, where an $\mathrm{L}^{\natural}$-convex function generalizes a submodular function so that the domain is a subset of the integer lattice. We propose computationally efficient algorithms for the online $\mathrm{L}^{\natural}$-convex function minimization in two major settings: the full information and the bandit settings. We analyze the regrets of these algorithms and show in particular that our algorithm for the full information setting obtains a tight regret bound up to a constant factor. We also demonstrate several motivating examples that illustrate the usefulness of the online $\mathrm{L}^{\natural}$-convex minimization.

cs.LG↗

arXiv2024

Fairness and efficiency trade-off in two-sided matching

The theory of two-sided matching has been extensively developed and applied to many real-life application domains. As the theory has been applied to increasingly diverse types of environments, researchers and practitioners have encountered various forms of distributional constraints. As a mechanism can handle a more general class of constraints, we can assign students more flexibly to colleges to increase students' welfare. However, it turns out that there exists a trade-off between students' welfare (efficiency) and fairness (which means no student has justified envy). Furthermore, this trade-off becomes sharper as the class of constraints becomes more general. The first contribution of this paper is to clarify the boundary on whether a strategyproof and fair mechanism can satisfy certain efficiency properties for each class of constraints. Our second contribution is to establish a weaker fairness requirement called envy-freeness up to $k$ peers (EF-$k$), which is inspired by a similar concept used in the fair division of indivisible items. EF-$k$ guarantees that each student has justified envy towards at most $k$ students. By varying $k$, EF-$k$ can represent different levels of fairness. We investigate theoretical properties associated with EF-$k$. Furthermore, we develop two contrasting strategyproof mechanisms that work for general hereditary constraints, i.e., one mechanism can guarantee a strong efficiency requirement, while the other can guarantee EF-$k$ for any fixed $k$. We evaluate the performance of these mechanisms through computer simulation.

cs.GT↗

arXiv2024

Characterizing the integer points in 2-decomposable polyhedra by closedness under operations

Characterizing the solution sets in a problem by closedness under operations is recognized as one of the key aspects of algorithm development, especially in constraint satisfaction. An example from the Boolean satisfiability problem is that the solution set of a Horn conjunctive normal form (CNF) is closed under the minimum operation, and this property implies that minimizing a nonnegative linear function over a Horn CNF can be done in polynomial time. In this paper, we focus on the set of integer points (vectors) in a polyhedron, and study the relation between these sets and closedness under operations from the viewpoint of 2-decomposability. By adding further conditions to the 2-decomposable polyhedra, we show that important classes of sets of integer vectors in polyhedra are characterized by 2-decomposability and closedness under certain operations, and in some classes, by closedness under operations alone. The most prominent result we show is that the set of integer vectors in a unit-two-variable-per-inequality polyhedron can be characterized by closedness under the median and directed discrete midpoint operations, each of these operations was independently considered in constraint satisfaction and discrete convex analysis.

cs.DS↗

arXiv2023

A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems

Linear programming (LP) problems with gainfree Leontief substitution systems have been intensively studied in economics and operations research, and include the feasibility problem of a class of Horn systems, which arises in, e.g., polyhedral combinatorics and logic. This subclass of LP problems admits a strongly polynomial time algorithm, where devising such an algorithm for general LP problems is one of the major theoretical open questions in mathematical optimization and computer science. Recently, much attention has been paid to devising certifying algorithms in software engineering, since those algorithms enable one to confirm the correctness of outputs of programs with simple computations. In this paper, we provide the first combinatorial (and strongly polynomial time) certifying algorithm for LP problems with gainfree Leontief substitution systems. As a by-product, we answer affirmatively an open question whether the feasibility problem of the class of Horn systems admits a combinatorial certifying algorithm.

cs.DS↗

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↗