Analyzing Residual Random Greedy for monotone submodular maximization

Abstract

Residual Random Greedy (RRGreedy) is a natural randomized version of the greedy algorithm for submodular maximization. It was introduced to address non-monotone submodular maximization [1] and plays an important role in the deterministic algorithm for monotone submodular maximization that beats the (1/2)-factor barrier [2]. In this work, we analyze RRGreedy for monotone submodular functions along two fronts: (1) For matroid constrained maximization of monotone submodular functions with bounded curvature α, we show that RRGreedy achieves a (1/(1+α))-approximation in the worst-case (i.e., irrespective of the randomness in the algorithm). In particular, this implies that it achieves a (1/2)-approximation in the worst-case (not just in expectation). (2) We generalize RRGreedy to k matroid intersection constraints and show that the generalization achieves a (1/(k+1))-approximation in expectation relative to the optimum value of the Lovász relaxation over the intersection of k matroid polytopes. Our results suggest that RRGreedy is at least as good as Greedy for matroid and matroid intersection constraints.