An analysis of the greedy algorithm for the submodular set covering problem

Abstract

We consider the problem: min\(\{ \mathop \Sigma \limits_{j \in s} f_j :z(S) = z(N),S \subseteqq N\} \) wherez is a nondecreasing submodular set function on a finite setN. Whenz is integer-valued andz(Ø)=0, it is shown that the value of a greedy heuristic solution never exceeds the optimal value by more than a factor\(H(\mathop {\max }\limits_j z(\{ j\} ))\) where\(H(d) = \sum\limits_{i = 1}^d {\frac{1}{i}} \).

This generalises earlier results of Dobson and others on the applications of the greedy algorithm to the integer covering problem: min {fy: Ayb, y ε {0, 1}} wherea ij ,b i } ≧ 0 are integer, and also includes the problem of finding a minimum weight basis in a matroid.

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Wolsey, L.A. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2, 385–393 (1982). https://doi.org/10.1007/BF02579435

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AMS subject classification (1980)

  • 68 C 05
  • 68 C 25
  • 90 C 10
  • 05 B 35