Approximation for maximizing monotone non-decreasing set functions with a greedy method

Abstract

We study the problem of maximizing a monotone non-decreasing function \(f\) subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if \(f\) is submodular, the greedy algorithm will find a solution with value at least \(\frac{1}{2}\) of the optimal value under a general matroid constraint and at least \(1-\frac{1}{e}\) of the optimal value under a uniform matroid \((\mathcal {M} = (X,\mathcal {I})\), \(\mathcal {I} = \{ S \subseteq X: |S| \le k\}\)) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least \(\frac{1}{1+\mu }\) of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where \(\mu = \alpha \), if \(0 \le \alpha \le 1\); \(\mu = \frac{\alpha ^K(1-\alpha ^K)}{K(1-\alpha )}\) if \(\alpha > 1\); here \(\alpha \) is a constant representing the “elemental curvature” of \(f\), and \(K\) is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a \(1 - (\frac{\alpha + \cdots + \alpha ^{k-1}}{1+\alpha + \cdots + \alpha ^{k-1}})^k\) approximation under a uniform matroid constraint. Under this unified \(\alpha \)-classification, submodular functions arise as the special case \(0 \le \alpha \le 1\).