Stable marriage and roommates problems with restricted edges: complexity and approximability

Cseh, Á. and Manlove, D. F. (2016) Stable marriage and roommates problems with restricted edges: complexity and approximability. Discrete Optimization, 20, pp. 62-89. (doi:10.1016/j.disopt.2016.03.002)

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Abstract

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs. Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Manlove, Dr David
Authors: Cseh, Á., and Manlove, D. F.
College/School:College of Science and Engineering > School of Computing Science
Journal Name:Discrete Optimization
Publisher:Elsevier
ISSN:1572-5286
Published Online:16 April 2016
Copyright Holders:Copyright © 2016 Elsevier B.V.
First Published:First published in Discrete Optimization 20: 62-89
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
607071Efficient Algorithms for Mechanism Design Without Monetary Transfer.David ManloveEngineering & Physical Sciences Research Council (EPSRC)EP/K010042/1COM - COMPUTING SCIENCE