Solutions for the Stable Roommates Problem with Payments

Abstract

The stable roommates problem with payments has as input a graph G = (V,E) with an edge weighting w: E → ℝ₊ and the problem is to find a stable solution. A solution is a matching M with a vector $p\in{\mathbb R}^V_+$ that satisfies p _u  + p _v  = w(uv) for all uv ∈ M and p _u  = 0 for all u unmatched in M. A solution is stable if it prevents blocking pairs, i.e., pairs of adjacent vertices u and v with p _u  + p _v  < w(uv). By pinpointing a relationship to the accessibility of the coalition structure core of matching games, we give a simple constructive proof for showing that every yes-instance of the stable roommates problem with payments allows a path of linear length that starts in an arbitrary unstable solution and that ends in a stable solution. This result generalizes a result of Chen, Fujishige and Yang for bipartite instances to general instances. We also show that the problems Blocking Pairs and Blocking Value, which are to find a solution with a minimum number of blocking pairs or a minimum total blocking value, respectively, are NP-hard. Finally, we prove that the variant of the first problem, in which the number of blocking pairs must be minimized with respect to some fixed matching, is NP-hard, whereas this variant of the second problem is polynomial-time solvable.