Improved Approximation Algorithm for k-level Uncapacitated Facility Location Problem (with Penalties)

Improved Approximation Algorithm for k-level Uncapacitated Facility Location Problem (with Penalties)

Byrka, Jaroslaw (University of Wroclaw)
Li, S. (TU Delft Discrete Mathematics and Optimization)
Rybicki, Bartosz (University of Wroclaw)

2014-11-04

We study the k-level uncapacitated facility location problem (k-level UFL) in which clients need to be connected with paths crossing open facilities of k types (levels). In this paper we first propose an approximation algorithm that for any constant k, in polynomial time, delivers solutions of cost at most α_k times OPT, where α_k is an increasing function of k, with lim_k→∞α_k=3. Our algorithm rounds a fractional solution to an extended LP formulation of the problem. The rounding builds upon the technique of iteratively rounding fractional solutions on trees (Garg, Konjevod, and Ravi SODA’98) originally used for the group Steiner tree problem. We improve the approximation ratio for k-level UFL for all k ≥ 3, in particular we obtain the ratio equal 2.02, 2.14, and 2.24 for k = 3,4, and 5. Second, we give a simple interpretation of the randomization process (Li ICALP’2011) for 1-level UFL in terms of solving an auxiliary (factor revealing) LP. Armed with this simple view point, we exercise the randomization on our algorithm for the k-level UFL. We further improve the approximation ratio for all k ≥ 3, obtaining 1.97, 2.09, and 2.19 for k = 3,4, and 5. Third, we extend our algorithm to the k-level UFL with penalties (k-level UFLWP), in which the setting is the same as k-level UFL except that the planner has the option to pay a penalty instead of connecting chosen clients.

Approximation algorithms
Facility location

http://resolver.tudelft.nl/uuid:0eebc93e-92e5-4e65-8307-a5d5e23b5238

https://doi.org/10.1007/s00224-014-9575-3

1432-4350

Theory of Computing Systems, 58 (1), 19-44

Institutional Repository

journal article

© 2014 Jaroslaw Byrka, S. Li, Bartosz Rybicki