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An analysis of approximations for maximizing submodular set functions—I

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LetN be a finite set andz be a real-valued 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 maxS⊂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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On leave of absence from Cornell University and supported, in part, by NSF Grant ENG 75-00568.

Supported, in part, by NSF Grant ENG 76-20274.

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Nemhauser, G.L., Wolsey, L.A. & Fisher, M.L. An analysis of approximations for maximizing submodular set functions—I. Mathematical Programming 14, 265–294 (1978).

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