Robust randomized matchings

Abstract

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. Hassin and Rubinstein [7] 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.

Scheduling When You Don't Know the Number of Machines

“…Another way of addressing uncertainty is to require one solution that is good against all possible values of the uncertain parameters. Examples of work in this direction include the universal traveling salesman problem (one tour that is good no matter which subset of points arrive) [15], robust matchings (one matching is chosen and then evaluated by its top k edges, where k is unknown) [9,13], a knapsack of unknown capacity(one policy of packing that is good irrespective of the actual capacity) [6] and 2-stage scheduling (some decisions must be made before the actual scenario is known) [5,17]. In scheduling problems, there are many ways to model uncertainty in the jobs, including online algorithms [1,2], in which the set of jobs is not known in advance, stochastic scheduling [12], in which the jobs are modeled as random variables, and work on schedules that are good against multiple objective functions [4,14,16].…”