 
Summary: Inapproximability of the Asymmetric
Facility Location and kMedian Problems
Aaron Archer #
April 22, 2000
Abstract
In the asymmetric versions of the uncapacitated facility location and kmedian problems,
distances satisfy the triangle inequality but the distances from point i to point j and from j to
i may di#er. For the facility location problem there is an O(log N) approximation algorithm
due to Hochbaum. For the kmedian problem, Lin and Vitter gave a bicriterion approximation
algorithm that blows up the service cost by a constant while blowing up the number of centers by
an O(log N) factor. Both of these algorithms are based on the greedy algorithm for the set cover
problem. In this note, we present inapproximability results for the asymmetric versions of the
uncapacitated facility location and kmedian problems, based on the hardness of approximating
set cover. Up to a constant, the above approximation algorithms are the best possible, unless
NP # DTIME(n O(log log n) ).
1 Introduction
The uncapacitated facility location (UFL) and kmedian problems are two central problems in the
study of clustering. Typically in these problems, one makes the assumption that the input points
lie in a metric space, which means that the distances c ij between points i and j are symmetric
(c ij = c ji ) and satisfy the triangle inequality (c ik # c ij +c jk ). Although the metric versions of these
