Greed is Not Always Good: On Submodular Maximization over Independence Systems
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In this work, we consider the maximization of submodular functions constrained by independence systems. Because of the wide applicability of submodular functions, this problem has been extensively studied in the literature. When the independence system is a p-system, prior literature has claimed that the greedy algorithm achieves a 1/(p+1)-approximation if the submodular function is monotone. We show that, on the contrary, for any ϵ > 0, the problem is hard to approximate within (2/n)^1-ϵ, where n is the size of the ground set, even when the independence system is a 1-system. This result invalidates prior work on constant-factor algorithms for non-monotone submodular maximization over p-systems as well. On the positive side, we provide the first nearly linear-time algorithm for maximization of non-monotone submodular functions over p-extendible independence systems, which are a subclass of p-systems.
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