- Maximizing Non-monotone Submodular Functions under Matroid and Knapsack Constraints Jon Lee (jonleeus.ibm.com) Vahab S. Mirrokni (mirroknigoogle.com) Viswanath Nagarajan (viswanatandrew.cmu.edu) Maxim Sviridenko (svirius.ibm.com) 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 $\left({1\over k+2+{1\over k}+\epsilon}\right)$-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 $\frac{1}{k+1}$. Keywords: submodular maximization, matroid, knapsack, approximation algorithm Category 1: Combinatorial Optimization Category 2: Combinatorial Optimization (Approximation Algorithms ) Category 3: Combinatorial Optimization (Graphs and Matroids ) Citation: IBM Research Report RC24679 Download: [PDF]Entry Submitted: 10/24/2008Entry Accepted: 10/25/2008Entry Last Modified: 10/27/2008Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.