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 $0 \le \alpha \le 1$.