Balanced stable marriage: How close is close enough?

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