# Maximizing Non-monotone Submodular Functions under Matroid and Knapsack Constraints

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 $\left({1\over k+2+{1\over k}+\epsilon}\right)$-approximation for the submodular maximization problem under $k$ matroid constraints, and a $\left({1\over 5}-\epsilon\right)$-approximation algorithm for this problem subject to $k$ knapsack constraints ($\epsilon>0$ is any constant). We improve the approximation guarantee of our algorithm to ${1\over k+1+{1\over k-1}+\epsilon}$ for $k\ge 2$ partition matroid constraints. This idea also gives a $\left({1\over k+\epsilon}\right)$-approximation for maximizing a monotone submodular function subject to $k\ge 2$ partition matroids, which improves over the previously best known guarantee of $\frac{1}{k+1}$.

## Citation

IBM Research Report RC24679