Solutions for the Stable Roommates Problem with Payments

The stable roommates problem with payments has as input a graph G = (V,E) with an edge weighting w : E → R≥0 and the problem is to find a stable solution. A solution is a matching M with a vector p ∈ R≥0 that satisfies pu + pv = w(uv) for all uv ∈ M and pu = 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 pu + pv < w(uv), or equivalently, if the total blocking value ∑ uv∈E max{0, w(uv)− (pu + pv)} = 0. By pinpointing a relationship to the accessibility of the coalition structure core of matching games, we give a 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 generalizes a result of Chen, Fujishige and Yang [2011] 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 NPhard. 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.