Robust randomized matchings

Abstract

The following zero-sum game is played on a weighted graph G: Alice selects a matching M in G and Bob selects a number k. Then, Alice receives a payoff equal to the ratio of the weight of the top k edges of M to optk, which is the maximum weight of a matching of size at most k in G. If M guarantees a payoff of at least a then it is called α-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a 1 / √2-robust matching, which is best possible for this setting. In this paper, we show that Alice can improve on the guarantee of 1 / √2 when allowing her to play a randomized strategy. For this setting, we devise a simple algorithm that returns a 1/ln(4)-robust randomized matching. The algorithm is based on the following non-trivial observation: If all edge weights are integer powers of 2, then any lexicographically optimum matching is 1-robust. We prove this property not only for matchings but for any independence system in which optk is a concave function of k. This class of systems includes ma-troid intersection, 6-matchings, and strong 2-exchange systems. We also show that our robustness results for randomized matchings translate to an asymptotic robustness guarantee for deterministic matchings: When restricting Bob's choice to cardinalities larger than a given constant, then Alice can find a single deterministic matching with approximately the same guaranteed payoff as in the randomized setting. In addition to the above results, we also give a new simple LP-based proof of Hassin and Rubinstein's original result.