Greed is Good: Near-Optimal Submodular Maximization via Greedy Optimization

Moran Feldman, Christopher Harshaw, Amin Karbasi

Abstract: It is known that greedy methods perform well for maximizing monotone submodular functions. At the same time, such methods perform poorly in the face of non-monotonicity. In this paper, we show - arguably, surprisingly - that invoking the classical greedy algorithm O(k√)-times leads to the (currently) fastest deterministic algorithm, called Repeated Greedy, for maximizing a general submodular function subject to k-independent system constraints. Repeated Greedy achieves (1+O(1/k√))k approximation using O(nrk√) function evaluations (here, n and r denote the size of the ground set and the maximum size of a feasible solution, respectively). We then show that by a careful sampling procedure, we can run the greedy algorithm only once and obtain the (currently) fastest randomized algorithm, called Sample Greedy, for maximizing a submodular function subject to k-extendible system constraints (a subclass of k-independent system constrains). Sample Greedy achieves (k+3)-approximation with only O(nr/k) function evaluations. Finally, we derive an almost matching lower bound, and show that no polynomial time algorithm can have an approximation ratio smaller than k+1/2−ε. To further support our theoretical results, we compare the performance of Repeated Greedy and Sample Greedy with prior art in a concrete application (movie recommendation). We consistently observe that while Sample Greedy achieves practically the same utility as the best baseline, it performs at least two orders of magnitude faster.