arXiv subjects

Yu Yokoi

Publications and source records attributed to Yu Yokoi.

30 records · Page 2Linked to original sources

arXiv2022

Incomplete List Setting of the Hospitals/Residents Problem with Maximally Satisfying Lower Quotas

To mitigate the imbalance in the number of assignees in the Hospitals/Residents problem, Goko et al. [Goko et al., Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties, Proc. STACS 2022, pp. 31:1--31:20] studied the Hospitals/Residents problem with lower quotas whose goal is to find a stable matching that satisfies lower quotas as much as possible. In their paper, preference lists are assumed to be complete, that is, the preference list of each resident (resp., hospital) is assumed to contain all the hospitals (resp., residents). In this paper, we study a more general model where preference lists may be incomplete. For four natural scenarios, we obtain maximum gaps of the best and worst solutions, approximability results, and inapproximability results.

cs.GT↗

arXiv2022

Hypergraph characterization of split matroids

We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being closed under duality and taking minors. Furthermore, these matroids proved to be useful in giving exact asymptotic bounds for the dimension of the Dressian, and also implied new results on the rays of the tropical Grassmannians. In the present paper, we introduce the notion of elementary split matroids, a subclass of split matroids that contains all connected split matroids. We give a hypergraph characterization of elementary split matroids in terms of independent sets, and show that the proposed class is closed not only under duality and taking minors but also truncation. We further show that, in contrast to split matroids, the proposed class can be characterized by a single forbidden minor. As an application, we provide a complete list of binary split matroids.

math.CO↗

arXiv2022

Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.

cs.GT↗

arXiv2022

Approximation by Lexicographically Maximal Solutions in Matching and Matroid Intersection Problems

We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly $2$. Furthermore, we prove that if the ratio is less than $2$, say $α$, then a lexicographically maximal solution achieves $(α/2)$-approximation, and this bound is tight.

math.CO↗

arXiv2021

An Approximation Algorithm for Maximum Stable Matching with Ties and Constraints

We present a polynomial-time $\frac{3}{2}$-approximation algorithm for the problem of finding a maximum-cardinality stable matching in a many-to-many matching model with ties and laminar constraints on both sides. We formulate our problem using a bipartite multigraph whose vertices are called workers and firms, and edges are called contracts. Our algorithm is described as the computation of a stable matching in an auxiliary instance, in which each contract is replaced with three of its copies and all agents have strict preferences on the copied contracts. The construction of this auxiliary instance is symmetric for the two sides, which facilitates a simple symmetric analysis. We use the notion of matroid-kernel for computation in the auxiliary instance and exploit the base-orderability of laminar matroids to show the approximation ratio. In a special case in which each worker is assigned at most one contract and each firm has a strict preference, our algorithm defines a $\frac{3}{2}$-approximation mechanism that is strategy-proof for workers.

cs.DS↗

arXiv2021

Fair and Truthful Mechanism with Limited Subsidy

The notion of \emph{envy-freeness} is a natural and intuitive fairness requirement in resource allocation. With indivisible goods, such fair allocations are unfortunately not guaranteed to exist. Classical works have avoided this issue by introducing an additional divisible resource, i.e., money, to subsidize envious agents. In this paper, we aim to design a truthful allocation mechanism of indivisible goods to achieve both fairness and efficiency criteria with a limited amount of subsidy. Following the work of Halpern and Shah, our central question is as follows: to what extent do we need to rely on the power of money to accomplish these objectives? For general valuations, the impossibility theorem of combinatorial auction translates to our setting: even if an arbitrarily large amount of money is available for use, no mechanism can achieve truthfulness, envy-freeness, and utilitarian optimality simultaneously when agents have general monotone submodular valuations. By contrast, we show that, when agents have matroidal valuations, there is a truthful allocation mechanism that achieves envy-freeness and utilitarian optimality by subsidizing each agent with at most $1$, the maximum marginal contribution of each item for each agent. The design of the mechanism rests crucially on the underlying matroidal M-convexity of the Lorenz dominating allocations.

cs.GT↗

arXiv2019

Subgame Perfect Equilibria of Sequential Matching Games

We study a decentralized matching market in which firms sequentially make offers to potential workers. For each offer, the worker can choose "accept" or "reject," but the decision is irrevocable. The acceptance of an offer guarantees her job at the firm, but it may also eliminate chances of better offers from other firms in the future. We formulate this market as a perfect-information extensive-form game played by the workers. Each instance of this game has a unique subgame perfect equilibrium (SPE), which does not necessarily lead to a stable matching and has some perplexing properties. We show a dichotomy result that characterizes the complexity of computing the SPE. The computation is tractable if each firm makes offers to at most two workers or each worker receives offers from at most two firms. In contrast, it is PSPACE-hard even if both firms and workers are related to at most three offers. We also study engineering aspects of this matching market. It is shown that, for any preference profile, we can design an offering schedule of firms so that the worker-optimal stable matching is realized in the SPE.

cs.GT↗

arXiv2019

Equitable Partitions into Matchings and Coverings in Mixed Graphs

Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings. In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching--covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the problems of partitioning a mixed graph into matching forests or mixed edge covers, so that all parts are equal with respect to some criterion, such as edge/arc numbers or total sizes. Moreover, we provide the best possible multicriteria equalization.

math.CO↗

arXiv2019

A Note on a Nearly Uniform Partition into Common Independent Sets of Two Matroids

The present note is a strengthening of a recent paper by K. Takazawa and Y. Yokoi (A generalized-polymatroid approach to disjoint common independent sets in two matroids, Discrete Mathematics (2019)). For given two matroids on $E$, under the same assumption in their paper to guarantee the existence of a partition of $E$ into $k$ common independent sets of the two matroids, we show that there exists a nearly uniform partition $\mathcal{P}$ of $E$ into $k$ common independent sets, where the difference of the cardinalities of any two sets in $\mathcal{P}$ is at most one.

math.CO↗

arXiv2019

A Generalized-Polymatroid Approach to Disjoint Common Independent Sets in Two Matroids

In this paper, we investigate the classes of matroid intersection admitting a solution for the problem of partitioning the ground set $E$ into $k$ common independent sets, where $E$ can be partitioned into $k$ independent sets in each of the two matroids. For this problem, we present a new approach building upon the generalized-polymatroid intersection theorem. We exhibit that this approach offers alternative proofs and unified understandings of previous results showing that the problem has a solution for the intersection of two laminar matroids and that of two matroids without $(k+1)$-spanned elements. Moreover, we newly show that the intersection of a laminar matroid and a matroid without $(k+1)$-spanned elements admits a solution. We also construct an example of a transversal matroid which is incompatible with the generalized-polymatroid approach.

math.CO↗

arXiv2017

Envy-free Matchings with Lower Quotas

While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas and not all doctor-hospital pairs are acceptable. We first show that, for an HR-LQ instance, we can efficiently decide the existence of an envy-free matching. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomial time if quotas are paramodular.

cs.GT↗

arXiv2017

List Supermodular Coloring with Shorter Lists

In 1995, Galvin proved that a bipartite graph $G$ admits a list edge coloring if every edge is assigned a color list of length $Δ(G)$, the maximum degree of the graph. This result was improved by Borodin, Kostochka and Woodall, who proved that $G$ still admits a list edge coloring if every edge $e=st$ is assigned a list of $\max\{d_{G}(s), d_{G}(t)\}$ colors. Recently, Iwata and Yokoi provided the list supermodular coloring theorem, that extends Galvin's result to the setting of Schrijver's supermodular coloring. This paper provides a common generalization of these two extensions of Galvin's result.

math.CO↗