An analysis of approximations for maximizing submodular set functions - I


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, 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

Extracted Key Phrases

Citations per Year

2,001 Citations

Semantic Scholar estimates that this publication has 2,001 citations based on the available data.

See our FAQ for additional information.

Cite this paper

@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} }