Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem

Abstract

For the problem maxlcub;Z(S): S is an independent set in the matroid Xrcub;, it is well-known that the greedy algorithm finds an optimal solution when Z is an additive set function (Rado-Edmonds theorem). Fisher, Nemhauser and Wolsey have shown that, when Z is a nondecreasing submodular set function satisfying Z(∅)=0, the greedy algorithm finds a solution with value at least half the optimum value. In this paper we show that it finds a solution with value at least 1/(1 + α) times the optimum value, where α is a parameter which represents the ‘total curvature’ of Z. This parameter satisfies 0≤α≤1 and α=0 if and only if the set function Z is additive. Thus the theorems of Rado-Edmonds and Fisher-Nemhauser-Wolsey are both contained in the bound 1/(1 + α). We show that this bound is best possible in terms of α. Another bound which generalizes the Rado-Edmonds theorem is given in terms of a ‘greedy curvature’ of the set function. Unlike the first bound, this bound can prove the optimality of the greedy algorithm even in instances where Z is not additive. A third bound, in terms of the rank and the girth of X, unifies and generalizes the bounds (e−1)/e known for uniform matroids and 12 for general matroids. We also analyze the performance of the greedy algorithm when X is an independence system instead of a matroid. Then we derive two bounds, both tight: The first one is [1−(1−α/K)k]/α where K and k are the sizes of the largest and smallest maximal independent sets in X respectively; the second one is 1/(p+α) where p is the minimum number of matroids that must be intersected to obtain X.