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. * A preliminary version of this paper appeared in the proceedings of SODA 2015 [15].

Randomized strategies for cardinality robustness in the knapsack problem

“…As we described in § 1.2, there exists a randomized strategy with robustness at least 1/ln 4 for bit-concave independence systems [11]. In this section, we show that there exists an instance of the knapsack problem for which no randomized strategy can achieve a constant robustness.…”

“…We address randomized strategies for the robust independence systems defined by an instance of the knapsack problem. Note that those independence systems are not necessarily bit-concave, and hence the method in [11] cannot be applied. In what follows, we assume that F ̸ = 2 E and {e} ∈ F for every e ∈ E. That is, w e ≤ C for every e ∈ E and w(E) > C.…”

“…The above results correspond to deterministic strategies (or pure strategies) of the zero-sum game. Matuschke, Skutella, and Soto [11] introduced randomized strategies (or mixed strategies) for the robust independence systems. In this setting, Alice calls a probability distribution on the feasible sets, and Bob, knowing the distribution of Alice, chooses an integer k. The robustness of Alice's strategy is defined by the expected payoff.…”

“…For the robust matching case, Matuschke, Skutella, and Soto [11] presented a randomized strategy with robustness 1/ ln 4, which shows that randomized strategies break the bound on the robustness 1/ √ 2 of the deterministic strategies. They further showed that this strategy attains robustness 1/ ln 4 for bitconcave independence systems.…”

General bounds for incremental maximization

“…Matuschke at al. [28] describe a randomized algorithm for this problem that, under the assumption that the adversary does not know the outcome of the randomness, has competitive ratio ln(4) ≈ 1.386.…”