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Best Algorithms for Approximating the Maximum of a Submodular Set Function

G. L. Nemhauser and L. A. Wolsey
Mathematics of Operations Research
Vol. 3, No. 3 (Aug., 1978), pp. 177-188
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3689488
Page Count: 12
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Best Algorithms for Approximating the Maximum of a Submodular Set Function
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Abstract

A real-valued function z whose domain is all of the subsets of N = {1,..., n} is said to be submodular if $z(S)+z(T)\geq z(S\cup T)+z(S\cap T),\forall S,T\subseteq N$, and nondecreasing if $z(S)\leq z(T),\forall S\subset T\subseteq N$. We consider the problem ${\rm max}_{S\subset N}\ \{z(S)\colon |S|\geq K$, z submodular and nondecreasing, z(φ) = 0}. Many combinatorial optimization problems can be posed in this framework. For example, a well-known location problem and the maximization of certain boolean polynomials are in this class. We present a family of algorithms that involve the partial enumeration of all sets of cardinality q and then a greedy selection of the remaining elements, q = 0,..., K - l. For fixed K, the qth member of this family requires $O(n^{q+1})$ computations and is guaranteed to achieve at least $[1-(\frac{K-q}{K})(\frac{K-q-1}{K-q})^{K-q}]\times 100$ percent of the optimum value. Our main result is that this is the best performance guarantee that can be obtained by any algorithm whose number of computations does not exceed $O(n^{q+1})$.

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