# An analysis of approximations for maximizing submodular set functions—I

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## 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(S⋃T)+z(S⋂T) for all*S, T* in*N.* 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 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 − 1)/e, where e is the base of the natural logarithm.

## Key words

Heuristics Greedy Algorithm Interchange Algorithm Linear Programming Matroid Optimization Submodular Set Functions## Preview

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