Inapproximability of Stable Marriage Problems

The stable marriage problem has received considerable attention both due to its practical applications as well as its mathematical structure. While the original problem has all participants rank all members of the opposite sex in a strict order of preference, two natural variations are to allow for incomplete preference lists and ties in the preferences. Both variations are polynomially solvable by a variation of the classical algorithm of Gale and Shapley. On the other hand, it has recently been shown to be NP-hard to nd a maximum cardinality stable matching when both of the variations are allowed. We show here that it is APX-hard to approximate the maximum cardinality stable matching with incomplete lists and ties. This holds for some very restricted instances both in terms of lengths of preference lists and lengths and occurrences of ties in the lists. We also obtain optimal (N) hardness results for 'egalitarian' and 'minimum regret' variants.