Randomized approximation of the stable marriage problem

While the original stable marriage problem requires all participants to rank all members of the opposite sex in a strict order, two natural variations are to allow for incomplete preference lists and ties in the preferences. Either variation is polynomially solvable, but it has recently been shown to be NP-hard to find a maximum cardinality stable matching when both of the variations are allowed. It is easy to see that the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. In this paper, we give a randomized approximation algorithm RANDBRK and show that its expected approximation ratio is at most 10/7(< 1.4286) for a restricted but still NP-hard case, where ties occur in only men's lists, each man writes at most one tie, and the length of ties is two. We also show that our analysis is nearly tight by giving a lower bound 32/23(> 1.3913) for RANDBRK. Furthermore, we show that these restrictions except for the last one can be removed without increasing the approximation ratio too much. (C) 2004 Elsevier B.V. All rights reserved.