Randomized Approximation for Generalized Median Stable Matching

This paper deals with finding a generalized median stable matching (GMSM), introducedby Teo and Sethuraman (1998) as a fair stable marriage. Cheng (2008) showed that findingthe i-th GMSM is #P-hard in case of i = O(N), where N is the number of stable matchingsof an instance. She also gave an exact algorithm running in polynomial time in case ofi = O(log logN), and the complexity remained as open in case of i is ω(log logN) and o(N).In this paper, we establish two hardness results. We show that finding the i-th GMSMis #P-hard even when i = O(N), where c ≥ 1 is an arbitrary constant, and that decidingif a matching can be a GMSM is #P-hard. On the other hand, we give a polynomial timeexact algorithm in case that i is O((logN) ′) where c′ is an arbitrary positive constant. Wealso propose two randomized approximation schemes for the i-th GMSM using an oracle foralmost uniformly sampling ideals of a partially ordered set (poset). This is the first resulton randomized approximation schemes for the GMSM.