Abstract
In this paper, we consider an extension of the classical facility location problem, namely k-facility location problem with linear penalties. In contrast to the classical facility location problem, this problem opens no more than k facilities and pays a penalty cost for any non-served client. We present a local search algorithm for this problem with a similar but more technical analysis due to the extra penalty cost, compared to that in Zhang (Theoretical Computer Science 384:126–135, 2007). We show that the approximation ratio of the local search algorithm is \(2 + 1/p + \sqrt{3+ 2/p+ 1/p^2} + \epsilon \), where \(p \in {\mathbb {Z}}_+\) is a parameter of the algorithm and \(\epsilon >0\) is a positive number.
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References
Archer A, Bateni M, Hajiaghayi M, Karloff H (2011) Improved approximation algorithms for prize-collecting Steiner tree and TSP. SIAM J Comput 40:309–332
Arya V, Garg N, Khandekar R, Meyerson A, Munagala K, Pandit V (2004) Local search heuristics for \(k\)-median and facility location problems. SIAM J Comput 33:544–562
Bienstock D, Goemans MX, Simchi-Levi D, Williamson D (1993) A note on the prize collecting traveling salesman problem. Math Program 59:413–420
Bartal Y, Leonardi S, Marchetti-Spaccamela A, Sgall J, Stougie L (2000) Multiprocessor scheduling with rejection. SIAM J Discret Math 13:64–78
Charikar M, Guha S (2005) Improved combinatorial algorithms for facility location problems. SIAM J Comput 34:803–824
Charikar M, Khuller S, Mount DM, Narasimhan G (2001) Algorithms for facility location problems with outliers. In: Proceedings of SODA, pp 642–651
Gupta N, Gupta S (2014) Approximation algorithms for capacitated facility location problem with penalties. CoRR, abs/1408.4944v4
Hajiaghayi M, Khandekar R, Kortsarz G (2012) Local search algorithms for the red-blue median problem. Algorithmica 63:795–814
Hochbaum DS (1982) Heuristics for the fixed cost median problem. Math Program 22:148–162
Jain K, Mahdian M, Saberi A (2002) A new greedy approach for facility location problems. In: Proceedings of STOC, pp 731–740
Jain K, Mahdian M, Markakis E, Saberi A, Vazirani VV (2003) Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J ACM 50:795–824
Jain K, Vazirani VV (2001) Approximation algorithms for metric facility location and \(k\)-median problems using the primal-dual schema and Lagrangian relaxation. J ACM 48:274–296
Li S (2013) A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf Comput 222:45–58
Li Y, Du D, Xiu N, Xu D (2015) Improved approximation algorithms for the facility location problems with linear/submodular penalties. Algorithmica 73:460–482
Shabtay D, Gaspar N, Kaspi M (2013) A survey on offline scheduling with rejection. J Sched 16:3–28
Wang Y, Xu D, Du D, Wu C (2015) Local search algorithms for k-median and k-facility location problems with linear penalties. In: Proceedings of COCOA, pp 60–71
Zhang P (2007) A new approximation algorithm for the \(k\)-facility location problem. Theor Comput Sci 384:126–135
Acknowledgments
The research of the second author is supported by NSFC (No. 11531014). The third author’s research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 06446. The fourth author’s research is supported by NSFC (No. 11501412).
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A preliminary version of this paper appeared in Proceedings of 9th Annual International Conference on Combinatorial Optimization and Applications, pp 60–71, 2015.
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Wang, Y., Xu, D., Du, D. et al. An approximation algorithm for k-facility location problem with linear penalties using local search scheme. J Comb Optim 36, 264–279 (2018). https://doi.org/10.1007/s10878-016-0080-2
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DOI: https://doi.org/10.1007/s10878-016-0080-2