  


Maximizing Nonmonotone Submodular Functions under Matroid and Knapsack Constraints
Jon Lee (jonleeus.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 NPhard. In this paper, we give the first constantfactor approximation algorithm for maximizing any nonnegative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for nonmonotone 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 ${1\over k+1+{1\over k1}+\epsilon}$ 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: 10/24/2008 Modify/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  