Connected facility location via random facility sampling and core detouring

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. The mentioned results can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems.

[1]  Luca Becchetti,et al.  Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy , 2005, SODA '05.

[2]  Fabrizio Grandoni,et al.  Improved Approximation for Single-Sink Buy-at-Bulk , 2006, ISAAC.

[3]  Friedrich Eisenbrand,et al.  An improved approximation algorithm for virtual private network design , 2005, SODA '05.

[4]  Aravind Srinivasan,et al.  Cost-Sharing Mechanisms for Network Design , 2007, Algorithmica.

[5]  David R. Karger,et al.  Building Steiner trees with incomplete global knowledge , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[6]  Said Salhi,et al.  Discrete Location Theory , 1991 .

[7]  Amit Kumar,et al.  Provisioning a virtual private network: a network design problem for multicommodity flow , 2001, STOC '01.

[8]  David P. Williamson,et al.  A simpler and better derandomization of an approximation algorithm for single source rent-or-buy , 2007, Oper. Res. Lett..

[9]  Lisa Zhang,et al.  The access network design problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[10]  R. Ravi,et al.  Boosted sampling: approximation algorithms for stochastic optimization , 2004, STOC '04.

[11]  Tim Roughgarden,et al.  Approximation via cost sharing , 2007, J. ACM.

[12]  R. Ravi,et al.  When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks , 1995, SIAM J. Comput..

[13]  Adam Meyerson,et al.  Online facility location , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[14]  S. E. Dreyfus,et al.  The steiner problem in graphs , 1971, Networks.

[15]  Bhaba R. Sarker,et al.  Discrete location theory , 1991 .

[16]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[17]  Tim Roughgarden,et al.  Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[18]  R. Ravi,et al.  Approximation Algorithms for the Traveling Purchaser Problem and its Variants in Network Design , 1999, ESA.

[19]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.

[20]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[21]  Balaji Raghavachari,et al.  Improved Approximation Algorithms for the Single-Sink Buy-at-Bulk Network Design Problems , 2004, SWAT.

[22]  Martin Skutella,et al.  New Approaches for Virtual Private Network Design , 2005, ICALP.

[23]  Kunal Talwar,et al.  The Single-Sink Buy-at-Bulk LP Has Constant Integrality Gap , 2002, IPCO.

[24]  Sudipto Guha,et al.  A Constant Factor Approximation for the Single Sink Edge Installation Problem , 2009, SIAM J. Comput..

[25]  Kyung-Yong Chwa,et al.  Improved Primal-Dual Approximation Algorithm for the Connected Facility Location Problem , 2008, COCOA.

[26]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[27]  Friedrich Eisenbrand,et al.  Approximating connected facility location problems via random facility sampling and core detouring , 2008, SODA '08.

[28]  Mohammad Mahdian,et al.  A 2-Approximation Algorithm for the Soft-Capacitated Facility Location Problem , 2003, RANDOM-APPROX.

[29]  Mohammad Mahdian,et al.  Improved Approximation Algorithms for Metric Facility Location Problems , 2002, APPROX.

[30]  J. Vygen.  Approximation Algorithms for Facility Location Problems ( Lecture Notes ) , 2005 .

[31]  Evangelos Markakis,et al.  Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP , 2002, JACM.

[32]  Mihalis Yannakakis,et al.  The Traveling Salesman Problem with Distances One and Two , 1993, Math. Oper. Res..

[33]  R. Ravi,et al.  When trees collide: an approximation algorithm for the generalized Steiner problem on networks , 1991, STOC '91.

[34]  Marshall W. Bern,et al.  The Steiner Problem with Edge Lengths 1 and 2 , 1989, Inf. Process. Lett..

[35]  Chaitanya Swamy,et al.  Primal-Dual Algorithms for Connected Facility Location Problems , 2002, APPROX.

[36]  Anke van Zuylen.  Deterministic Sampling Algorithms for Network Design , 2008, ESA.

[37]  Jochen Könemann,et al.  Simple cost sharing schemes for multicommodity rent-or-buy and stochastic Steiner tree , 2006, STOC '06.

[38]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[39]  Chaitanya Swamy,et al.  Primal–Dual Algorithms for Connected Facility Location Problems , 2004, Algorithmica.

[40]  Sudipto Guha,et al.  A constant factor approximation for the single sink edge installation problems , 2001, STOC '01.

[41]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

[42]  R. Ravi,et al.  Approximation Algorithms for Problems Combining Facility Location and Network Design , 2006, Oper. Res..

[43]  Alex Zelikovsky,et al.  Improved Steiner tree approximation in graphs , 2000, SODA '00.