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Journal of Global Optimization

, Volume 75, Issue 3, pp 833–849 | Cite as

Parametric monotone function maximization with matroid constraints

  • Suning Gong
  • Qingqin NongEmail author
  • Wenjing Liu
  • Qizhi Fang
Article
  • 119 Downloads

Abstract

We study the problem of maximizing an increasing function \(f:2^N\rightarrow \mathcal {R}_{+}\) subject to matroid constraints. Gruia Calinescu, Chandra Chekuri, Martin Pál and Jan Vondrák have shown that, if f is nondecreasing and submodular, the continuous greedy algorithm and pipage rounding technique can be combined to find a solution with value at least \(1-1/e\) of the optimal value. But pipage rounding technique have strong requirement for submodularity. Chandra Chekuri, Jan Vondrák and Rico Zenklusen proposed a rounding technique called contention resolution schemes. They showed that if f is submodular, the objective value of the integral solution rounding by the contention resolution schemes is at least \(1-1/e\) times of the value of the fractional solution. Let \(f:2^N\rightarrow \mathcal {R}_{+}\) be an increasing function with generic submodularity ratio \(\gamma \in (0,1]\), and let \((N,\mathcal {I})\) be a matroid. In this paper, we consider the problem \(\max _{S\in \mathcal {I}}f(S)\) and provide a \(\gamma (1-e^{-1})(1-e^{-\gamma }-o(1))\)-approximation algorithm. Our main tools are the continuous greedy algorithm and contention resolution schemes which are the first time applied to nonsubmodular functions.

Keywords

Increasing set function Matroid Generic submodularity ratio Approximation algorithm 

Notes

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical ScienceOcean University of ChinaQingdaoPeople’s Republic of China

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