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

Cseh, Á. and Manlove, D. (2015) Stable marriage and roommates problems with restricted edges: complexity and approximability. Lecture Notes in Computer Science, 9347, pp. 15-26. (doi:10.1007/978-3-662-48433-3_2)

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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. [5] 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 on restricted pairs. Our main theorems prove that for the (bipartite) stable marriage problem, case (1) leads to NP-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite stable roommates instances, case (2) yields an NP-hard but (under some cardinality assumptions) 2-approximable problem. In the case of NP-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
Additional Information:Presented at SAGT 2015: the 8th International Symposium on Algorithmic Game Theory, Saarbrücken, Germany, 28- 30 Sept. 2015.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Manlove, Dr David
Authors: Cseh, Á., and Manlove, D.
College/School:College of Science and Engineering > School of Computing Science
Journal Name:Lecture Notes in Computer Science
Publisher:Springer
ISSN:0302-9743
ISSN (Online):1611-3349
Copyright Holders:Copyright © 2015 The Authors
First Published:First published in Lecture Notes in Computer Science 9347:15-26
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher
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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