Improved Combinatorial Algorithms for the Facility Location and k-Median Problems

We present improved combinatorial approximation algorithms for the uncapacitated facility location and k-median problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414+ in ~ O(n= ) time. This also yields a bicriteria approximation tradeo of (1+ ; 1+ 2= ) for facility cost versus service cost which is better than previously known tradeo s and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to Jain and Vazirani, we get an approximation ratio of 1:853 in ~ O(n) time. This is already very close to the approximation guarantee of the best known algorithm which is LP-based. Further, combined with the best known LP-based algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1:728. We present improved approximation algorithms for a variant of the capacitated facility location problem. We also present a 4-approximation for the k-median problem, building on and improving the recent 6-approximation due to Jain and Vazirani. The algorithm runs in ~ O(n) time. [email protected]. Stanford University, Stanford, CA 94305. Research supported by the Pierre and Christine Lamond Fellowship, NSF Grant IIS-9811904 and NSF Award CCR-9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. [email protected]. Stanford University, Stanford, CA 94305. Research Supported by IBM Cooperative Fellowship, NSF Grant IIS-9811904 and NSF Award CCR-9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.