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 α 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.
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