Constrained Stable Marriage with Free Edges or Few Blocking Pairs

Yinghui Wen, Jiong Guo

Abstract: Given two disjoint sets U and W, where the members (also called agents) of U and W are called men and women, respectively, and each agent is associated with an ordered preference list that ranks a subset of the agents from the opposite gender, a stable matching is a set of pairwise disjoint woman-man pairs admitting no blocking pairs. A blocking pair refers to a woman and a man, who prefer each other to their partners in the matching. Gale and Shapley proved that a stable matching exists for every instance. Since then, a lot of stable matching variants have been introduced. For instance, the \(\pi \)-stable marriage problem asks for a stable matching satisfying a given constraint \(\pi \). Unlike in the unconstrained case, a given instance may not admit a \(\pi \)-stable matching and one has to accept a semi-stable matching satisfying \(\pi \), for instance, a matching satisfying \(\pi \) and admitting few blocking pairs. In this paper, we study two such problems, namely, \(\pi \) -Stable Marriage with Free Edges (\(\pi \)-SMFE) and \(\pi \) -Stable Marriage with \(t\) -Blocking Pairs (\(\pi \)-SM\(t\)BP). \(\pi \)-SMFE seeks for a matching M satisfying \(\pi \) and the condition that all blocking pairs occurring in M are from a given set F of woman-man pairs, while the solution matchings of \(\pi \)-SM\(t\)BP need to satisfy \(\pi \) and admit at most t blocking pairs. We examine four constraints, Regret, Egalitarian, Forced, and Forbidden, and prove that both \(\pi \)-SMFE and \(\pi \)-SM\(t\)BP are NP-hard for all four constraints even with complete preference lists. Concerning parameterized complexity, we establish a series of fixed-parameter tractable and intractable results for \(\pi \)-SMFE and \(\pi \)-SM\(t\)BP with respect to some structural parameters such as the number of agents and the number of free edges/blocking pairs.