Hostname: page-component-77c78cf97d-sp94z Total loading time: 0 Render date: 2026-04-24T12:36:18.369Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Average Case Performance of Some Greedy Approximation Algorithms For the Uncapacitated Facility Location Problem

Published online by Cambridge University Press:  01 September 2007

ABRAHAM D. FLAXMAN ,
ALAN M. FRIEZE  and
JUAN CARLOS VERA
ABRAHAM D. FLAXMAN
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA (e-mail: abie@cmu.edu, alan@random.math.cmu.edu, jvera@andrew.cmu.edu)
ALAN M. FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA (e-mail: abie@cmu.edu, alan@random.math.cmu.edu, jvera@andrew.cmu.edu)
JUAN CARLOS VERA
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA (e-mail: abie@cmu.edu, alan@random.math.cmu.edu, jvera@andrew.cmu.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

In combinatorial optimization, a popular approach to NP-hard problems is the design of approximation algorithms. These algorithms typically run in polynomial time and are guaranteed to produce a solution which is within a known multiplicative factor of optimal. Unfortunately, the known factor is often known to be large in pathological instances. Conventional wisdom holds that, in practice, approximation algorithms will produce solutions closer to optimal than their proven guarantees. In this paper, we use the rigorous-analysis-of-heuristics framework to investigate this conventional wisdom.

We analyse the performance of three related approximation algorithms for the uncapacitated facility location problem (from Jain, Mahdian, Markakis, Saberi and Vazirani (2003) and Mahdian, Ye and Zhang (2002)) when each is applied to an instances created by placing n points uniformly at random in the unit square. We find that, with high probability, these 3 algorithms do not find asymptotically optimal solutions, and, also with high probability, a simple plane partitioning heuristic does find an asymptotically optimal solution.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 5 , September 2007 , pp. 713 - 732
Copyright
Copyright © Cambridge University Press 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Ahn, S., Cooper, C., Cornuéjols, G. and Frieze, A. M. (1988) Probabilistic analysis of a relaxation for the k-median problem. Math. Oper. Res. 13 1–31.CrossRefGoogle Scholar
[2]Balinski, M. L. (1965) Integer programming: Methods, uses, computation. Management Science 12 253–313.CrossRefGoogle Scholar
[3]Barahona, F. and Chudak, F. A. (1999) Solving large scale uncapacitated facility location problems. In Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, Kluwer Academic, pp. 48–62.Google Scholar
[4]Charikar, M. and Guha, S. (1999) Improved combinatorial algorithms for facility location and k-median problems. In Proc. 40th Annual IEEE Symposium on Foundations of Computer Science, pp. 378–388.CrossRefGoogle Scholar
[5]Chudak, F. A. and Shmoys, D. B.Improved approximation algorithms for the uncapacitated facility location problem. SIAM J. Comput., Vol. 33, Issue 1, pp. 1–25 (2004).CrossRefGoogle Scholar
[6]Coffman, E. G. Jr., Courcoubetis, C., Garey, M. R., Johnson, D. S., Shor, P. W., Weber, R. R. and Yannakakis, M. (2002) Perfect packing theorems and the average-case behaviour of optimal and online bin packing. SIAM Review 44 95–108.CrossRefGoogle Scholar
[7]Coja-Oghlan, A., Krumke, S. O. and Nierhoff, T. (2003) A heuristic for the Stacker Crane Problem on trees which is almost surely exact. In Algorithms and Computation: 14th International Symposium, ISAAC 2003, Kyoto, Vol. 2906 of Lecture Notes in Computer Science, pp. 605–614CrossRefGoogle Scholar
[8]Cornuéjols, G., Nemhauser, G. L. and Wolsey, L. A. (1990) The uncapacitated facility location problem. In Discrete Location Theory, Wiley, pp. 119–171.Google Scholar
[9]Feige, U. and Kilian, J. (2001) Heuristics for semirandom graph problems. J. Comput. System Sci. 63 639–673.CrossRefGoogle Scholar
[10]Hoefer, M. (2003) Experimental comparison of heuristic and approximation algorithms for uncapacitated facility location. In Proc. 2nd Intl. Workshop on Experimental and Efficient Algorithms (WEA 2003), Vol. 2647 of Lecture Notes in Computer Science, Springer, pp. 165–178.CrossRefGoogle Scholar
[11]Jain, K., Mahdian, M., Markakis, E., Saberi, A. and Vazirani, V. V. (2003) Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. Assoc. Comput. Mach. 50 795–824.Google Scholar
[12]Jain, K. and Vazirani, V. V. (1999) Primal–dual approximation algorithms for metric facility location and k-median problems. In Proc. 40th Annual IEEE Symposium on Foundations of Computer Science, pp. 2–13.CrossRefGoogle Scholar
[13]Korupolu, M. R., Plaxton, C. G. and Rajaraman, R. (1998) Analysis of a local search heuristic for facility location problems. In Proc. 9th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 1–10.Google Scholar
[14]Krarup, J. and Pruzan, P. M. (1983) The simple plant location problem: Survey and synthesis. European J. Oper. Research 12 36–81.CrossRefGoogle Scholar
[15]Mahdian, M., Ye, Y. and Zhang, J. (2002) Improved approximation algorithms for metric facility location problems. In Proc. 5th Intl. Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 229–242.Google Scholar
[16]Penrose, M. (2003) Random Geometric Graphs, Oxford University Press.CrossRefGoogle Scholar
[17]Shmoys, D. B., Tardos, E. and Aardal, K. I. (1997) Approximation algorithms for facility location problems. In Proc. 29th Annual ACM Symposium on the Theory of Computation, pp. 265–274.Google Scholar
[18]Steele, J. M. (1997) Probability Theory and Combinatorial Optimization, Vol. 69 of CBMS–NSF Regional Conference Series in Applied Mathematics, SIAM.Google Scholar
[19]Sviridenko, M. (2002) An improved approximation algorithm for the metric uncapacitated facility location problem. In Proc. Conference on Integer Programming and Combinatorial Optimization (IPCO02), Vol. 2337 of Lecture Notes in Computer Science, Springer, pp. 230–239.Google Scholar
[20]Yukich, J. E. (1998) Probability Theory of Classical Euclidean Optimization Problems, Springer.Google Scholar