An Approximation Algorithm for Maximum Stable Matching with Ties and Constraints

We present a polynomial-time 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 3/2-approximation mechanism that is strategy-proof for workers

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. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. 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

Solving the Maximum Popular Matching Problem with Matroid Constraints

We consider the problem of finding a maximum popular matching in a many-to-many matching setting with two-sided preferences and matroid constraints. This problem was proposed by Kamiyama (2020) and solved in the special case where matroids are base orderable. Utilizing a newly shown matroid exchange property, we show that the problem is tractable for arbitrary matroids. We further investigate a different notion of popularity, where the agents vote with respect to lexicographic preferences, and show that both existence and verification problems become NP-hard, even in the $b$-matching case.Comment: 16 pages, 2 figure

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

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.Comment: 12 pages, 1 figur

マトロイド的構造における安定マッチング

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 岩田 覚, 東京大学教授 今井 浩, 東京大学教授 定兼 邦彦, 東京大学准教授 平井 広志, 慶應義塾大学教授 田村 明久University of Tokyo(東京大学

A Museum Exhibits Support System Based on History and Culture Literacy

[[abstract]]Museums need an interactive data collection and visualisation tool for their artefacts. This paper describes a study in which we enable access to Chinese and Japanese cultural heritage information from two history museums, the National Palace Museum in Taiwan and the Tokugawa Art Museum in Japan. Results from these museum databases were used to develop a prototype system to demonstrate advanced cultural learning and historical timeline functionalities for foreigners. This system is based on temporal data from the museums’ databases, and provided the user with powerful data manipulation and graphical visualisation tools. It might become a basis of an interactive digital museum system for Chinese and Japanese heritages especially for foreign users.[[journaltype]]國外[[ispeerreviewed]]Y[[booktype]]電子版[[countrycodes]]GB