Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. [7] gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution A preliminary version of this paper appeared in the Proceedings of SAGT 2015: the 8th International Symposium on Algorithmic Game Theory. Email addresses: [email protected] (Ágnes Cseh), [email protected] (David F. Manlove) Supported by COST Action IC1205 on Computational Social Choice and by the Deutsche Telekom Stiftung. Part of this work was carried out whilst visiting the University of Glasgow. Present address: School of Computer Science, Reykjavik University, Menntavegur 1, 101 Reykjavik, Iceland. Supported by Engineering and Physical Sciences Research Council grant EP/K010042/1. Preprint submitted to Discrete Optimization April 19, 2016 to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints on restricted pairs. Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to NP-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an NP-hard but (under some cardinality assumptions) 2-approximable problem. In the case of NP-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs.