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Lagrangian relaxation for the kmedian problem: new insights and continuity properties

Summary: Lagrangian relaxation for the k­median problem:
new insights and continuity properties
Aaron Archer # , Ranjithkumar Rajagopalan ## , and David B. Shmoys # # #
School of Operations Research and Industrial Engineering, Cornell University
Ithaca, NY 14853 Email:{aarcher,ranjith,shmoys}@cs.cornell.edu
Abstract. This work gives new insight into two well­known approximation algo­
rithms for the uncapacitated facility location problem: the primal­dual algorithm
of Jain &Vazirani, and an algorithm of Mettu & Plaxton. Our main result answers
positively a question posed by Jain & Vazirani of whether their algorithm can be
modified to attain a desired ``continuity'' property. This yields an upper bound of
3 on the integrality gap of the natural LP relaxation of the k­median problem, but
our approach does not yield a polynomial time algorithm with this guarantee. We
also give a new simple proof of the performance guarantee of the Mettu­Plaxton
algorithm using LP duality, which suggests a minor modification of the algorithm
that makes it Lagrangian­multiplier preserving.
1 Introduction
Facility location problems have been widely studied in both the operations research
and computer science literature We consider the two most popular variants of facility
location: the k­median problem and the uncapacitated facility location problem (UFL).
In both cases, we are given a set C of clients who must be served by a set F of facilities,


Source: Archer, Aaron - Algorithms and Optimization Group, AT&T Labs-Research


Collections: Computer Technologies and Information Sciences