 
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,
