Journal of the Operations Research Society of Japan

MANY-TO-MANY STABLE MATCHINGS WITH TIES IN TREES

Keita Nakamura, Naoyuki Kamiyama

2016 Volume 59 Issue 3 Pages 225-240
Published: 2016
Released on J-STAGE: July 11, 2016

DOIhttps://doi.org/10.15807/jorsj.59.225

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In the stable matching problem introduced by Gale and Shapley, it is known that in the case where the preference lists may involve ties, a stable matching always exists, but the sizes of stable matchings may be different. In this paper, we consider the problem of finding a maximum-size stable matching in a many-to-many matching market with ties. It is known that this problem is NP-hard even if the capacity of every agent is one. In this paper, we prove that this problem in trees can be solved in polynomial time by extending the algorithm proposed by Tayu and Ueno for the one-to-one setting.

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A STRUCTURAL GEOMETRICAL ANALYSIS OF WEAKLY INFEASIBLE SDPS

Bruno F. Lourenço, Masakazu Muramatsu, Takashi Tsuchiya

2016 Volume 59 Issue 3 Pages 241-257
Published: 2016
Released on J-STAGE: July 11, 2016

DOIhttps://doi.org/10.15807/jorsj.59.241

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In this article, we develop a detailed analysis of semidefinite feasibility problems (SDFPs) to understand how weak infeasibility arises in semidefinite programming. This is done by decomposing a SDFP into smaller problems, in a way that preserves most feasibility properties of the original problem. The decomposition utilizes a set of vectors (computed in the primal space) which arises when we apply the facial reduction algorithm to the dual feasible region. In particular, we show that for a weakly infeasible problem over n× n matrices, at most n-1 directions are required to approach the positive semidefinite cone. We also present a discussion on feasibility certificates for SDFPs and related complexity results.

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A NEW METHOD OF APPROXIMATION FOR COMPUTING SYSTEM-FAILURE FREQUENCY

Masahiro Hayashi, Hisao Yamamoto

2016 Volume 59 Issue 3 Pages 258-268
Published: 2016
Released on J-STAGE: July 11, 2016

DOIhttps://doi.org/10.15807/jorsj.59.258

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A method for approximately computing the frequency of system failures is proposed. The method uses multilinear polynomials to compute lower and upper bounds of system availability and transforms these bounds into lower and upper bounds of the system failure frequency without increasing the computational complexity. Most of the currently used high-speed approximation methods that compute the lower and upper bounds of availability use multilinear polynomials, so it is a relatively simple matter to transform these methods into new ones for computing the failure frequency while preserving their speed and accuracy. Experimental results demonstrate that the proposed method can quickly and accurately compute the system failure frequency.

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