# Improved approximation algorithms for capacitated facility location problems

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## Abstract.

In a surprising result, Korupolu, Plaxton, and Rajaraman [13] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8+*ε* of the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces a solution which is within a factor of 6(1+*ε*) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron.

Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn any *α*-approximation algorithm for this variant into a polynomial-time algorithm which, at an additional cost of twice the optimum of the standard CFLP, opens at most one additional copy of any facility. This allows us to transform a recent 2-approximation algorithm of Mahdian, Ye, and Zhang [17] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than four times the value of the standard CFLP.

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### References

- 1.Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993Google Scholar
- 2.Arora, S., Sudan, M.: Improved low-degree testing and its applications. In: Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 485–495Google Scholar
- 3.Babayev, Dj.A.: Comments on the note of Frieze. Math. Program.
**7**, 249–252 (1974)Google Scholar - 4.Barahona, F., Jensen, D.: Plant location with minimum inventory. Math. Program.
**83**, 101–111 (1998)Google Scholar - 5.Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and
*k*-median problems. In: Proceedings of the 40th IEEE Symposium on the Foundations of Computer Science, 1999, pp. 378–388Google Scholar - 6.Chudak, F.: Improved approximation algorithms for uncapacitated facility location. In: Proceedings of the 6th IPCO Conference, 1998, pp. 180–194Google Scholar
- 7.Chudak, F., Shmoys, D.B.: Improved approximation algorithms for the uncapacitated facility location problem. SIAM J. Comput.
**33**, 1–25 (2003)Google Scholar - 8.Chudak, F., Shmoys, D.B.: Improved approximation algorithms for a capacitated facility location problem. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999, pp. 875–876Google Scholar
- 9.Chudak, F., Williamson, D.P.: Improved approximation algorithms for capacitated facility location problems. In: Proceedings of the 7th IPCO Conference, 1999, pp. 99–113Google Scholar
- 10.Cornuéjols, G., Nemhauser, G., Wolsey, L.: The uncapacitated facility location problem. In: P. Mirchandani and R. Francis, (eds.), Discrete Location Theory, John Wiley and Sons, Inc., New York, 1990, pp. 119–171Google Scholar
- 11.
- 12.Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. J. Algorithms
**31**, 228–248 (1999)Google Scholar - 13.Korupolu, M., Plaxton, C., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. J. Algorithms
**37**, 146–188 (2000)Google Scholar - 14.Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, New York, 1976Google Scholar
- 15.Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM
**41**, 960–981 (1994)CrossRefMATHGoogle Scholar - 16.Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, 2002, pp. 229–242Google Scholar
- 17.Mahdian, M., Ye, Y., Zhang, J.: A 2-approximation algorithm for the soft-capacitated facility location problem. In: Proceedings of the 6th International Workshop on Approximation Algorithms for Combinatorial Optimization, 2003, pp. 129–140Google Scholar
- 18.Mirchandani, P., Francis, R., eds.: Discrete Location Theory. John Wiley and Sons, Inc., New York, 1990Google Scholar
- 19.Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions – I. Math. Program.
**14**, 265–294 (1978)Google Scholar - 20.Pál, M., Tardos, É., Wexler, T.: Facility location with nonuniform hard capacities. In: Proceedings of the 41st IEEE Symposium on the Foundations of Computing, 2001, pp. 329–338Google Scholar
- 21.Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 475–484Google Scholar
- 22.Shmoys, D., Tardos, É, Aardal, K.: Approximation algorithms for facility location problems. In: Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 265–274Google Scholar
- 23.Sviridenko, M.: Personal communication, July, 1998Google Scholar