Non-monotone submodular maximization under matroid and knapsack constraints

TLDR

Submodular function maximization is a central NP‑hard combinatorial optimization problem that generalizes many classic problems such as Max Cut, constraint satisfaction, maximum‑entropy sampling, and facility location. The authors aim to provide the first constant‑factor approximation algorithm for maximizing any non‑negative submodular function under multiple matroid or knapsack constraints. They design a constant‑factor approximation algorithm that works for non‑monotone submodular functions and applies to arbitrary collections of matroid or knapsack constraints. The algorithm achieves a (1/(k+2+1/k+ε))‑approximation for k matroid constraints, a (1/5−ε)‑approximation for k knapsack constraints, improves to 1/(k+1+{1/k−1}+ε) for k≥2 partition matroids, and gives a (1/(k+ε))‑approximation for monotone submodular functions under k≥2 partition matroids, surpassing the previous 1/(k+1) bound.

Abstract

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a (1/k+2+1/k+ε)-approximation for the submodular maximization problem under k matroid constraints, and a (1/5-ε)-approximation algorithm for this problem subject to k knapsack constraints (ε>0 is any constant). We improve the approximation guarantee of our algorithm to 1/k+1+{1/k-1}+ε for k≥2 partition matroid constraints. This idea also gives a ({1/k+ε)-approximation for maximizing a monotone submodular function subject to k≥2 partition matroids, which improves over the previously best known guarantee of 1/k+1.