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, \(\mathcal {A}(a)\), who a ranks strictly; we use \(p_{a}(b)\) to denote the position of \(b\in \mathcal{A}(a)\) in a’s preference list. The objective is to decide whether there exists a stable matching \(\mu \) such that \(\mathsf{balance}(\mu )\triangleq \max \{\sum _{(m,w)\in \mu }p_{m}(w), \sum _{(m,w)\in \mu } p_{w}(m)\}\le k\). In SM, all stable matchings match the same set of agents, \(A^\star \) which can be computed in polynomial time. As \(\mathsf{balance}(\mu )\ge \frac{|A^\star |}{2}\) for any stable matching \(\mu \), BSM is trivially fixed-parameter tractable (FPT) with respect to k. Thus, a natural question is whether BSM is FPT with respect to \(k-\frac{|A^\star |}{2}\). 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 \(k-\frac{|A^\star |}{2}\)). The two extreme stable matchings are the man-optimal \(\mu _M\) and the woman-optimal \(\mu _W\). Let \(O_{M}=\sum _{(m,w)\in \mu _M} p_m(w)\), and \(O_{W}= \sum _{(m,w)\in \mu _W} p_w(m)\). In this work, we prove that BSM parameterized by \(t = k - \min \{O_{M},O_{W}\}\) 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 \(t = k - \max \{O_{M},O_{W}\}\) is W[1]-hard.