Maximizing Nonmonotone Submodular Functions under Matroid or 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. Un...

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Vydáno v:	SIAM journal on discrete mathematics Ročník 23; číslo 4; s. 2053 - 2078
Hlavní autoři:	Lee, Jon, Mirrokni, Vahab S., Nagarajan, Viswanath, Sviridenko, Maxim
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Témata:	Algebra Algorithmics. Computability. Computer arithmetics Algorithms Applied mathematics Applied sciences Approximation Combinatorics Combinatorics. Ordered structures Computer science; control theory; systems Convex and discrete geometry Design of experiments Entropy Exact sciences and technology Geometry Graph theory Graphs Mathematical programming Mathematics Optimization Random variables Sciences and techniques of general use Theoretical computing Hypergraph Constraint Combinatorial problem Matroid Optimization method Entropy 68W25 approximation algorithms Approximation algorithm Combinatorial optimization Location problem Maximum matroid and knapsack constraints Discrete mathematics Sampling Monotonic function Partition Approximation Maximization Minimization 90C27 Knapsack problem submodular maximization Graph cut Digraph Directed graph
ISSN:	0895-4801, 1095-7146
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Popis
Shrnutí:	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 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 $\frac{1}{k+1+\frac{1}{k-1}+\epsilon}$ 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 $\frac{1}{k+1}$.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14
ISSN:	0895-4801 1095-7146
DOI:	10.1137/090750020