Skip to main content Accessibility help
×
Home
Hostname: page-component-78bd46657c-2pqp7 Total loading time: 0.252 Render date: 2021-05-07T17:18:24.819Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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
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)
Corresponding

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.

Type
Paper
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.

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. 13 131.CrossRefGoogle Scholar
[2]Balinski, M. L. (1965) Integer programming: Methods, uses, computation. Management Science 12 253313.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.Google 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. 125 (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 95108.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. 605–614Google Scholar
[8]Cornuéjols, G., Nemhauser, G. L. and Wolsey, L. A. (1990) The uncapacitated facility location problem. In Discrete Location Theory, Wiley, pp. 119171.Google Scholar
[9]Feige, U. and Kilian, J. (2001) Heuristics for semirandom graph problems. J. Comput. System Sci. 63 639673.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. 165178.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 795824.CrossRefGoogle 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.Google 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 3681.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. 229242.CrossRefGoogle 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. 265274.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. 230239.Google Scholar
[20]Yukich, J. E. (1998) Probability Theory of Classical Euclidean Optimization Problems, Springer.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *