Greedy is Good: Constrained Non-submodular Function Maximization via Weak Submodularity

The widely used greedy algorithm has been recently shown to achieve near-optimal theoretical guarantees for the problems of constrained monotone non-submodular function maximization, with competitive performances in practice. In this paper, we investigate the problems of maximizing monotone non-submodular set functions under three classes of independent system constraints, including p-matroid intersection constraints, p-extendible system constraints and p-system constraints. We prove that the greedy algorithm yields an approximation ratio of gamma/p+gamma for the former two problems, and xi gamma/p+xi gamma for the last problem, which further has been improved to gamma/p+gamma where gamma,xi denote the submodularity ratio and the diminishing returns ratio of set function respectively. In addition, we also show that the greedy guarantees have a further refinement of xi/p+alpha xi for all the problems mentioned above, where a is the generalized curvature alpha of set function. Finally, we show that our greedy algorithm does yield competitive practical performances using a variety of experiments on synthetic data.