Balanced stable marriage: How close is close enough?

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), whom a ranks strictly; we use p_a(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(μ)≜max⁡{∑_(m,w)∈μp_m(w),∑_(m,w)∈μp_w(m)}≤k. In SM, all stable matchings match the same set of agents, A^⋆ which can be computed in polynomial time. As [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 μ_M and the woman-optimal μ_W. Let O_M=∑_(m,w)∈μMp_m(w), and O_W=∑_(m,w)∈μWp_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.