LP Rounding for k-Centers with Non-uniform Hard Capacities
release_5esduijwuzc4bfdsjzwg4cicpa
by
Marek Cygan, MohammadTaghi Hajiaghayi, Samir Khuller
2012
Abstract
In this paper we consider a generalization of the classical k-center problem
with capacities. Our goal is to select k centers in a graph, and assign each
node to a nearby center, so that we respect the capacity constraints on
centers. The objective is to minimize the maximum distance a node has to travel
to get to its assigned center. This problem is NP-hard, even when centers have
no capacity restrictions and optimal factor 2 approximation algorithms are
known. With capacities, when all centers have identical capacities, a 6
approximation is known with no better lower bounds than for the infinite
capacity version.
While many generalizations and variations of this problem have been studied
extensively, no progress was made on the capacitated version for a general
capacity function. We develop the first constant factor approximation algorithm
for this problem. Our algorithm uses an LP rounding approach to solve this
problem, and works for the case of non-uniform hard capacities, when multiple
copies of a node may not be chosen and can be extended to the case when there
is a hard bound on the number of copies of a node that may be selected. In
addition we establish a lower bound on the integrality gap of 7(5) for
non-uniform (uniform) hard capacities. In addition we prove that if there is a
(3-eps)-factor approximation for this problem then P=NP.
Finally, for non-uniform soft capacities we present a much simpler
11-approximation algorithm, which we find as one more evidence that hard
capacities are much harder to deal with.
In text/plain
format
Archived Files and Locations
application/pdf 517.4 kB
file_wzegpnxv4vgzbcs7rftavrmgeq
|
arxiv.org (repository) web.archive.org (webarchive) |
1208.3054v1
access all versions, variants, and formats of this works (eg, pre-prints)