An analysis of approximations for maximizing submodular set functions - I

Abstract

Let N be a finite set and z be a real-valued function defined on the set of subsets of N that satisfies z(S)+z(T)>-z(SUT)+z(SnT) for all S, T in N. Such a function is called submodular. We consider the problem maXscN {z(S): IS[ <-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 than K 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, when z(S) is nondecreasing and z(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 each K and has a limiting value of ( e l)/e, where e is the base of the natural logarithm.

DOI: 10.1007/BF01588971

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@article{Nemhauser1978AnAO, title={An analysis of approximations for maximizing submodular set functions - I}, author={George L. Nemhauser and Laurence A. Wolsey and Marshall L. Fisher}, journal={Math. Program.}, year={1978}, volume={14}, pages={265-294} }