Efficient algorithms for generalized Stable Marriage and Roommates problems

Fleiner, T., Irving, R.W. and Manlove, D.F. (2007) Efficient algorithms for generalized Stable Marriage and Roommates problems. Theoretical Computer Science, 381(1-3), pp. 162-176. (doi: 10.1016/j.tcs.2007.04.029)

Abstract

We consider a generalization of the Stable Roommates problem (SR), in which preference lists may be partially ordered and forbidden pairs may be present, denoted by SRPF. This includes, as a special case, a corresponding generalization of the classical Stable Marriage problem (SM), denoted by SMPF. By extending previous work of Feder, we give a two-step reduction from SRPF to 2-SAT. This has many consequences, including fast algorithms for a range of problems associated with finding "optimal" stable matchings and listing all solutions, given variants of SR and SM. For example, given an SMPF instance I, we show that there exists an O(m) "succinct" certificate for the unsolvability of I, an O(m) algorithm for finding all the super-stable pairs in I, an O(m+kn) algorithm for listing all the super-stable matchings in I, an O(m^1.5) algorithm for finding an egalitarian super-stable matching in I, and an O(m) algorithm for finding a minimum regret super-stable matching in I, where n is the number of men, m is the total length of the preference lists, and k is the number of super-stable matchings in I. Analogous results apply in the case of SRPF.

• Item Type: Articles
• Keywords: Stable Roommates problem; Stable Marriage problem; Partial order; Forbidden pair; Super-stable matching
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Manlove, Professor David
• Authors: Fleiner, T., Irving, R.W., and Manlove, D.F.
• Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science; Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Computing Science
• Research Group: Formal Analysis, Theory and Algorithms
• Journal Name: Theoretical Computer Science
• Publisher: Elsevier
• ISSN: 0304-3975
• Copyright Holders: Copyright © 2007 Elsevier
• First Published: First published in Theoretical Computer Science 381(1-3):162-176
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher