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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Bibliographic Details
Published in:SIAM journal on discrete mathematics Vol. 23; no. 4; pp. 2053 - 2078
Main Authors: Lee, Jon, Mirrokni, Vahab S., Nagarajan, Viswanath, Sviridenko, Maxim
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Subjects:
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
Online Access:Get full text
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Summary: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 $\frac{1}{k+1}$.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:0895-4801
1095-7146
DOI:10.1137/090750020