## 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.

This is a preview of subscription content, log in to check access.

## References

- 1.
Aardal, K.I., Chudak, F.A., Shmoys, D.B.: A \(3\)-approximation algorithm for the \(k\)-level uncapacitated facility location problem. Info. Process. Lett.

**72**, 161–167 (1999) - 2.
Ageev, A., Ye, Y., Zhang, J.: Improved combinatorial approximation algorithms for the \(k\)-level facility location problem. SIAM J. Discrete Math.

**18**, 207–217 (2003) - 3.
Byrka, J., Li, S., Rybicki, B.: Improved approximation algorithm for \(k\)-level UFL with penalties. In: Proceedings of the 11th Workshop on Approximation and Online Algorithms (WAOA), pp. 85–96 (2013)

- 4.
Byrka, J., Aardal, K.I.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM J. Comput.

**39**, 2212–2231 (2010) - 5.
Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and \(k\)-median problems. In: Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pp. 378–388 (1999)

- 6.
Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of the 12th Annual Symposium on Discrete Algorithms (SODA), pp. 642–651 (2001)

- 7.
Chen, X., Chen, B.: Approximation algorithms for soft-capacitated facility location in capacitated network design. Algorithmica

**53**, 263–297 (2007) - 8.
Chudak, F.A., Nagano, K.: Efficient solutions to relaxations of combinatorial problems with submodular penalties via the Lovász extension and non-smooth convex optimization. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 79–88 (2007)

- 9.
Chudak, F.A., Shmoys, D.B.: Improved approximation algorithms for the uncapacitated facility location problem. SIAM J. Comput.

**33**, 1–25 (2003) - 10.
Du, D., Lu, R., Xu, D.: A primal-dual approximation algorithm for the facility location problem with submodular penalties. Algorithmica

**63**, 191–200 (2012) - 11.
Fujishige, S.: Submodular Functions and Optimization. Annals of Discrete Mathematics, 2nd edn. Elsevier, Amsterdam (2005)

- 12.
Geunes, J., Levi, R., Romeijn, H.E., Shmoys, D.B.: Approximation algorithms for supply chain planning and logistics problems with market choice. Math. Program.

**130**, 85–106 (2011) - 13.
Guha, S., Khuller, S.: Greedy strike back: improved facility location algorithms. J. Algorithms

**31**, 228–248 (1999) - 14.
Hayrapetyan, A., Swamy, C., Tardös, E.: Network design for information networks. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 933–942 (2005)

- 15.
Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: Proceedings on 34th Annual ACM Symposium on Theory of Computing (STOC), pp. 731–740 (2002)

- 16.
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and \(k\)-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM

**48**, 274–296 (2001) - 17.
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM

**50**, 795–824 (2003) - 18.
Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1–10 (1998)

- 19.
Li, Y., Du, D., Xiu, N., Xu, D.: Improved approximation algorithms for the facility location problems with linear/submodular penalty. In: Proceedings of the 19th Annual International Computing and Combinatorics Conference (COCOON), pp. 292–303 (2013)

- 20.
Li, Y., Du, D., Xiu, N., Xu, D.: A combinatorial \(2.375\)-approximation algorithm for the facility location problem with submodular penalties. Theoret. Comput. Sci.

**476**, 109–117 (2013) - 21.
Li, S.: A \(1.488\) approximation algorithm for the uncapacitated facility location problem. Info. Comput.

**222**, 45–58 (2013) - 22.
Mahdian, M.: Facility location and the analysis of algorithms through factor-revealing programs. Ph. D. thesis, MIT, Cambridge, MA (2004)

- 23.
Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. SIAM J. Comput.

**36**, 411–432 (2006) - 24.
Shmoys, D.B., Tardös, E., Aardal, K.I.: Approximation algorithms for facility location problems. In: Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (STOC), pp. 265–274 (1997)

- 25.
Shmoys, D.B., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM

**53**, 978–1012 (2006) - 26.
Shu, J., Teo, C.P., Shen, Z.J.: Max: stochastic transportation-inventory network design problem. Oper. Res.

**53**, 48–60 (2005) - 27.
Sviridenko, M.: An improved approximation algorithm for the metric uncapacitated facility location problem. In: Proceedings of 9th International Integer Programming and Combinatorial Optimization Conference (IPCO), pp. 240–257 (2002)

- 28.
Vygen, J.: Approximation algorithms for facility location problems (Lecture Notes). Report No. 05950-OR, Research Institute for Discrete Mathematics, University of Bonn, http://www.or.uni-bonn.de/vygen/fl. Accessed (2005)

- 29.
Xu, G., Xu, J.: An LP rounding algorithm for approximating uncapacitated facility location problem with penalties. Info. Process. Lett.

**94**, 119–123 (2005) - 30.
Xu, G., Xu, J.: An improved approximation algorithm for uncapacitated facility location problem with penalties. J. Comb. Optim.

**17**, 424–436 (2008) - 31.
Ye, Y., Zhang, J.: An approximation algorithm for the dynamic facility location problem. Combinatorial Optimization in Communication Networks, pp. 623–637. Kluwer Academic Publishers, Norwell MA (2005)

- 32.
Zhang, J., Chen, B., Ye, Y.: A multiexchange local search algorithm for the capacitated facility location problem. Math. Oper. Res.

**30**, 389–403 (2005) - 33.
Zhang, J.: Approximating the two-level facility location problem via a quasi-greedy approach. Math. Program.

**108**, 159–176 (2006) - 34.
Zhang, P.: A new approximation algorithm for the \(k\)-facility location problem. Theoret. Comput. Sci.

**384**, 126–135 (2007)

## 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.

## Author information

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

Issue Date:

### Keywords

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