An improved approximation algorithm for the stable marriage problem with one-sided ties
Paper in proceeding, 2014

We consider the problem of computing a large stable matching in a bipartite graph G = (A ∪ B, E) where each vertex u εA ∪ B ranks its neighbors in an order of preference, perhaps involving ties. A matching M is said to be stable if there is no edge (a,b) such that a is unmatched or prefers b to M(a) and similarly, b is unmatched or prefers a to M(b). While a stable matching in G can be easily computed in linear time by the Gale-Shapley algorithm, it is known that computing a maximum size stable matching is APX-hard. In this paper we consider the case when the preference lists of vertices in A are strict while the preference lists of vertices in B may include ties. This case is also APX-hard and the current best approximation ratio known here is 25/17 ≈ 1.4706 which relies on solving an LP. We improve this ratio to 22/15 ≈ 1.4667 by a simple linear time algorithm. We first compute a half-integral stable matching in {0,0.5,1}|E| and round it to an integral stable matching M. The ratio |OPT|/|M| is bounded via a payment scheme that charges other components in OPT ⊕ M to cover the costs of length-5 augmenting paths. There will be no length-3 augmenting paths here. We also consider the following special case of two-sided ties, where every tie length is 2. This case is known to be UGC-hard to approximate to within 4/3. We show a 10/7 ≈ 1.4286 approximation algorithm here that runs in linear time.

Stable marriage problem

Bipartite graphs

Stable matching

Payment schemes

Approximation algorithms

Gale-shapley algorithms

Combinatorial optimization

Integer programming

Best approximations

Linear-time algorithms

Preference lists


Chien-Chung Huang

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

T. Kavitha

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

03029743 (ISSN) 16113349 (eISSN)

Vol. 8494 297-308
978-3-319-07556-3 (ISBN)

Subject Categories

Computer and Information Science





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