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Approximation algorithm for the k-level capacitated facility location problem

Approximation algorithm for the k-level capacitated facility location problem. Xing Wang Beijing University of Technology Joint work with Donglei Du, Dachuan Xu. outline. K-CFLP Special cases Algorithm Result Discussion. k-level capacitated facility location problem(k-CFLP).

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Approximation algorithm for the k-level capacitated facility location problem

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  1. Approximation algorithm for the k-level capacitated facility location problem Xing Wang Beijing University of Technology Joint work with Donglei Du, Dachuan Xu

  2. outline • K-CFLP • Special cases • Algorithm • Result • Discussion

  3. k-level capacitated facility location problem(k-CFLP) • Given a complete (k+1) bipartite graph are disjoint sets, are facilities,each , is specified by a cost and a capacity , is client set ,each client has a demand

  4. The objective is to open some facilities on each level and to connect each client to a path along open facilities such that the total open and connection cost is minimized.

  5. The (metric) uncapacitated facility location problem (UFLP) • Shmoys et al. -3.16 (the first constant approximation factor) -[1997], Proceedings of STOC • Byrka and Aardal -1.50 (the current best approximation factor ) -to appear in SIAM Journal on Computing

  6. The (metric) k-level uncapacitated facility location problem (k-FLP) • Aardal et al. 3 LP rounding algorithm [1999], Information Processing Letters 72 • Zhang et al. 1.77 LP rounding algorithm (2-FLP) [2006], Mathematical Programming 108 • Ageev et al. 3.27 best combinatorial algorithm [2005], SIAM Journal on Discrete Mathematics

  7. The (metric) 1-level capacitated facility location problem (1-CFLP) • When the capacities are uniform - Korupolu et al. 8 local search algorithm [1998], Proceedings of SODA - Chudak and Williamson 5.83 local search algorithm [1999], Proceedings of IPOC

  8. The (metric) 1-level capacitated facility location problem (1-CFLP) • When the capacities are not uniform -Pal et al. 8.53 local search algorithm [2001], Proceedings of FOCS -Mahdian and Pal 7.88 local search algorithm [2003], Proceedings of ESA - Zhang et al. (currently best) multiexchange local search (MLS) algorithm [2005], Mathematics of Operations Research

  9. The (metric) 1-level capacitated facility location problem (1-CFLP) When all facility opening costs are equal -Levi et al. 5 (first upper bound on the integrality gap) LP rounding algorithm [2004], Proceedings of IPCO

  10. Algorithm • For any instance M of the k-CFLP, we construct -an instance of 1-CFLP -an instance of (k-1)-CFLP

  11. Step 1. Solve the instance of 1-CFLP by the MLS algorithm • Step 2. Solve recursively the instance of (k-1)-CFLP • Step 3 .Constructing a transportation problem for each client • Step 4 .Constructing a solution for instance M

  12. Result • TheoremLet k 2, for any solution SOL of M, the solution ALG restrieved by Algorithm satisfies

  13. CorollaryFor any constant ,algorithm runs in polynomial time with approximation guarantee of

  14. Discussions • Further improve the overall approximation factor for the k-CFLP. • Give a lower bound on the polynomial time approximation factor k-CFLP instead of using the lower bound of UFLP as a special case.

  15. Thanks!

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