A Matroid Generalization of the Stable Matching Polytope

Fleiner, Tamás

doi:10.1007/3-540-45535-3_9

Cited by 32 publications

(48 citation statements)

References 7 publications

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“…Since the advent of Gale and Shapley's paper a large number of variations and extensions have been proposed in the literature. Recently, a remarkable extension has been made by Fleiner [10]. He extended the stable marriage model to the framework of matroids and showed the existence of a stable solution.…”

Section: §4 Overview Of Two-sided Matching Market Modelsmentioning

confidence: 99%

Applications of Discrete Convex Analysis to Mathematical Economics

Tamura

2004

Publ. Res. Inst. Math. Sci.

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Discrete convex analysis, which is a unified framework of discrete optimization, is being recognized as a basic tool for mathematical economics. This paper surveys the recent progress in applications of discrete convex analysis to mathematical economics. §1. Introduction Discrete convex analysis, proposed by Murota [25,26], is a unified framework of discrete optimization. Recently, applications of discrete convex analysis to mathematical economics have been investigated. The aim of this paper is to survey the following recent progress on this topic.The concepts of M-convex functions due to Murota [25,26] and M -convex functions due to Murota and Shioura [30], which play central roles in discrete convex analysis, are being recognized as nice discrete convex functions from the point of view of mathematical economics. For instance, for set functions, Fujishige and Yang [15] showed that M -concavity is equivalent to the gross substitutability and the single improvement property which are equivalent to each other for set functions [17] and are nice in the following sense. These properties guarantee the existence of the core of several models, e.g., a matching model proposed by Kelso and Crawford [21]. Relations among these three properties were extended to the general case by Danilov, Koshevoy and Lang [3] and by

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Section: §4 Overview Of Two-sided Matching Market Modelsmentioning

confidence: 99%

Applications of Discrete Convex Analysis to Mathematical Economics

Tamura

2004

Publ. Res. Inst. Math. Sci.

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“…In this paper, we propose a matroidbased approach [3,4,5] to the LCSM problem and we obtain the following results. Section 2, we describe our model.…”

Section: Introductionmentioning

confidence: 99%

A Matroid Approach to Stable Matchings with Lower Quotas

Fleiner¹,

Kamiyama²

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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In SODA'10, Huang introduced the laminar classified stable matching problem (LCSM for short) that is motivated by academic hiring. This problem is an extension of the wellknown hospitals/residents problem in which a hospital has laminar classes of residents and it sets lower and upper bounds on the number of residents that it would hire in that class. Against the intuition that stable matching problems with lower quotas are difficult in general, Huang proved that this problem can be solved in polynomial time. In this paper, we propose a matroid-based approach to this problem and we obtain the following results. (i) We solve a generalization of the LCSM problem. (ii) We exhibit a polyhedral description for stable assignments of the LCSM problem, which gives a positive answer to Huang's question. (iii) We prove that the set of stable assignments of the LCSM problem has a lattice structure similarly to the ordinary stable matching model.

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“…Related Works Our framework includes two major generalizations of the stable matching model: stable allocations in bipartite networks due to Baïou and Balinski [2] and matroid kernels in matroid intersection due to Fleiner [7,8]. On the other hand, it can be regarded as a special case of the discrete-concave function model of Eguchi, Fujishige, and Tamura [6].…”

Section: Our Contributionmentioning

confidence: 99%

“…We now describe the matroid kernel model of Fleiner [7,8]. Consider a pair of matroids (E, I H ) and (E, I F ) on the same ground set E with independent set families I H and I F .…”

Section: Our Contributionmentioning

confidence: 99%

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Finding a Stable Allocation in Polymatroid Intersection

Iwata¹,

Yokoi²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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The stable matching model of Gale and Shapley (1962) has been generalized in various directions such as matroid kernels due to Fleiner (2001) and stable allocations in bipartite networks due to Baïou and Balinski (2002). Unifying these generalizations, we introduce the concept of stable allocations in polymatroid intersection.Our framework includes both integer-and real-variable versions. The integer-variable version corresponds to a special case of the discrete-concave function model due to Eguchi, Fujishige, and Tamura (2003), who established the existence of a stable allocation by showing that a simple extension of the deferred acceptance algorithm of Gale and Shapley finds a stable allocation in pseudo-polynomial time.It has been open to develop a polynomial-time algorithm even for our special case.In this paper, we present the first strongly polynomial algorithm for finding a stable allocation in polymatroid intersection. To achieve this, we utilize the augmenting path technique for polymatroid intersection. In each iteration, the algorithm searches for an augmenting path by simulating a chain of proposes and rejects in the deferred acceptance algorithm. The running time of our algorithm is O(n 3 γ), where n and γ respectively denote the cardinality of the ground set and the time for computing the saturation and exchange capacities. This is as fast as the best known algorithm for the polymatroid intersection problem.

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A Matroid Generalization of the Stable Matching Polytope

Cited by 32 publications

References 7 publications

Applications of Discrete Convex Analysis to Mathematical Economics

Applications of Discrete Convex Analysis to Mathematical Economics

A Matroid Approach to Stable Matchings with Lower Quotas

Finding a Stable Allocation in Polymatroid Intersection

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