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Journal of Combinatorial Optimization

, Volume 31, Issue 1, pp 29–43 | Cite as

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

  • Zengfu Wang
  • Bill Moran
  • Xuezhi Wang
  • Quan Pan
Article

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 \(0 \le \alpha \le 1\).

Keywords

Monotone submodular set function Matroid Approximation algorithm 

Notes

Acknowledgments

This study was supported in part by the NSFC (No.61135001) and the AFOSR grant (FA2386-13-1-4080).

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Zengfu Wang
    • 1
    • 3
  • Bill Moran
    • 2
  • Xuezhi Wang
    • 2
  • Quan Pan
    • 1
  1. 1.School of AutomationNorthwestern Polytechnical UniversityXi’an China
  2. 2.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia
  3. 3.Department of Electrical and Electronic EngineeringThe University of MelbourneMelbourneAustralia

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