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Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

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Abstract

We consider the facility location problem with submodular penalties (FLPSP) and the facility location problem with linear penalties (FLPLP), two extensions of the classical facility location problem (FLP). First, we introduce a general algorithmic framework for a class of covering problems with submodular penalties, extending the recent result of Geunes et al. (Math Program 130:85–106, 2011) with linear penalties. This framework leverages existing approximation results for the original covering problems to obtain corresponding results for their counterparts with submodular penalties. Specifically, any LP-based \(\alpha \)-approximation for the original covering problem can be leveraged to obtain an \(\left( 1-e^{-1/\alpha }\right) ^{-1}\)-approximation algorithm for the counterpart with submodular penalties. Consequently, any LP-based approximation algorithm for the classical FLP (as a covering problem) can yield, via this framework, an approximation algorithm for the counterpart with submodular penalties. Second, by exploiting some special properties of submodular/linear penalty function, we present an LP rounding algorithm which has the currently best \(2\)-approximation ratio over the previously best \(2.375\) by Li et al. (Theoret Comput Sci 476:109–117, 2013) for the FLPSP and another LP-rounding algorithm which has the currently best \(1.5148\)-approximation ratio over the previously best \(1.853\) by Xu and Xu (J Comb Optim 17:424–436, 2008) for the FLPLP, respectively.

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Acknowledgments

The authors would like to thank the three anonymous referees for their valuable comments which greatly improved the manuscript. Particularly, one of the referees points out a key difference between Algorithm 4.2 and that in Li [21], which leads to a clarification on the definitions for various distances and a complete proof for Lemma 4.3. The fourth author thank Fengmin Wang, Yishui Wang, and Chenchen Wu for helpful discussions. This work was partially done while the first author was a visiting doctorate student at the Faculty of Business Administration, University of New Brunswick and supported in part by NSERC Grants 283103. The research of the second author is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 283106. The third author’s research is supported by the National Basic Research Program of China (No. 2010CB732501). The fourth author’s research is supported by NSF of China (Nos. 11071268, 11371001) and China Scholarship Council.

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Correspondence to Dachuan Xu.

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Li, Y., Du, D., Xiu, N. et al. Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties. Algorithmica 73, 460–482 (2015). https://doi.org/10.1007/s00453-014-9911-7

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Keywords

  • Approximation algorithm
  • Facility location problem
  • LP rounding
  • Submodular function