Balanced stable marriage: How close is close enough?

Abstract

Balanced Stable Marriage (BSM) is a central optimization version of the classic Stable Marriage (SM) problem. We study BSM from the viewpoint of Parameterized Complexity. Informally, the input of BSM consists of n men, n women, and an integer k. Each person a has a (sub)set of acceptable partners, A(a), who a ranks strictly; we use pa(b) to denote the position of b∈ A(a) in a’s preference list. The objective is to decide whether there exists a stable matching μ such that balance(formula presented). In SM, all stable matchings match the same set of agents, A* which can be computed in polynomial time. As balance (formula presented) for any stable matching μ, BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to (formula presented). With this viewpoint in mind, we draw a line between tractability and intractability in relation to the target value. This line separates additional natural parameterizations higher/lower than ours (e.g., we automatically resolve the parameterization (formula presented). The two extreme stable matchings are the man-optimal μM and the woman-optimal μW. Let (formula presented), and (formula presented). In this work, we prove that BSM parameterized by (formula presented) admits (1) a kernel where the number of people is linear in t, and (2) a parameterized algorithm whose running time is single exponential in t.BSM parameterized by (formula presented) is W[1]-hard.