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Maximizing nonmonotone submodular functions under matroid or knapsack constraints
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Lee, Jon, Mirrokni, Vahab S., Nagarajan, Viswanath and Sviridenko, Maxim (2010) Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM Journal on Discrete Mathematics, Vol.23 (No.4). pp. 20532078. doi:10.1137/090750020
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Official URL: http://dx.doi.org/10.1137/090750020
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 $(\frac{1}{k+2+\frac{1}{k}+\epsilon})$approximation for the submodular maximization problem under $k$ matroid constraints, and a $(\frac{1}{5}\epsilon)$approximation algorithm for this problem subject to $k$ knapsack constraints ($\epsilon>0$ is any constant). We improve the approximation guarantee of our algorithm to $\frac{1}{k+1+\frac{1}{k1}+\epsilon}$ for $k\geq2$ partition matroid constraints. This idea also gives a $(\frac{1}{k+\epsilon})$approximation for maximizing a monotone submodular function subject to $k\geq2$ partition matroids, which is an improvement over the previously best known guarantee of $\frac{1}{k+1}$.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software  
Divisions:  Faculty of Science > Computer Science  
Journal or Publication Title:  SIAM Journal on Discrete Mathematics  
Publisher:  Society for Industrial and Applied Mathematics  
ISSN:  08954801  
Official Date:  2010  
Dates: 


Volume:  Vol.23  
Number:  No.4  
Page Range:  pp. 20532078  
DOI:  10.1137/090750020  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access 
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