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 \(0 \le \alpha \le 1\).
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Notes
Given a set \(S \in \mathcal {I}\), return \(f(S)\).
Given an independence family \(\mathcal {I}\) and a set \(Y \subseteq X\), let \(\mathcal {B}(Y)\) be the set of maximal independent sets of \(\mathcal {I}\) included in \(Y\). Then \(\mathcal {I}\) is a \(p\)-system if, for all \(Y \subseteq X\), \(\frac{\max _{A \in \mathcal {B}(Y)} |A|}{\min _{A \in \mathcal {B}(Y)} |A|} \le p\). See the definition in Korte and Hausmann (1998) and Calinescu et al. (2011).
Given prices \(p_1,\ldots , p_n\), return a bundle \(S \in \arg \max _{T,T \subseteq X} f(T) - \sum _{i \in T} p_i\).
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This study was supported in part by the NSFC (No.61135001) and the AFOSR grant (FA2386-13-1-4080).
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Wang, Z., Moran, B., Wang, X. et al. Approximation for maximizing monotone non-decreasing set functions with a greedy method. J Comb Optim 31, 29–43 (2016). https://doi.org/10.1007/s10878-014-9707-3
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DOI: https://doi.org/10.1007/s10878-014-9707-3