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+/spl epsiv/ in O/spl tilde/(n/sup 2///spl epsiv/) time. This also yields a bicriteria approximation tradeoff of (1+/spl gamma/, 1+2//spl gamma/) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to K. Jain and V. Vazirani (1999), we get an approximation ratio of 1.853 in O/spl tilde/(n/sup 3/) 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 capacitated facility location and a variant. We also present a 4-approximation for the k-median problem, using similar ideas, building on the 6-approximation of Jain and Vazirani. The algorithm runs in O/spl tilde/(n/sup 3/) time.