Improved approximation algorithms for two variants of the stable marriage problem with ties

We consider the problem of computing a large stable matching in a bipartite graph G = (A∪ B, E) where each vertex u ∈ A∪ B ranks its neighbors in an order of preference, perhaps involving ties. Let the matched partner of u in a matching M be M ( u ). 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 first 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. Here we first compute a half-integral stable matching in {0,0.5,1}^|E| and then 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 next 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.