An analysis of approximations for maximizing submodular set functions—I
 G. L. Nemhauser,
 L. A. Wolsey,
 M. L. Fisher
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LetN be a finite set andz be a realvalued function defined on the set of subsets ofN that satisfies z(S)+z(T)≥z(S⋃T)+z(S⋂T) for allS, T inN. Such a function is called submodular. We consider the problem max_{S⊂N}{a(S):S≤K,z(S) submodular}.
Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.
We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1)/K] ^{ K } times the optimal value. This bound can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.
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 Title
 An analysis of approximations for maximizing submodular set functions—I
 Journal

Mathematical Programming
Volume 14, Issue 1 , pp 265294
 Cover Date
 19781201
 DOI
 10.1007/BF01588971
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Heuristics
 Greedy Algorithm
 Interchange Algorithm
 Linear Programming
 Matroid Optimization
 Submodular Set Functions
 Industry Sectors
 Authors

 G. L. Nemhauser ^{(1)}
 L. A. Wolsey ^{(1)}
 M. L. Fisher ^{(2)}
 Author Affiliations

 1. Center for Operations Research and Econometrics, University of Louvain, LouvainLaNeuve, Belgium
 2. University of Pennsylvania, Philadelphia, PA, USA