Robust Randomized Matchings

The following game is played on a weighted graph: Alice selects a matching M and Bob selects a number k. Alice's payoff is the ratio of the weight of the k heaviest edges of M to the maximum weight of a matching of size at most k. If M guarantees a payoff of at least a then it is called alpha-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a 1/root 2 -robust matching, which is best possible. We show that Alice can improve her payoff to 1/ln(4) by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.