23 research outputs found
Centrality of Trees for Capacitated k-Center
There is a large discrepancy in our understanding of uncapacitated and
capacitated versions of network location problems. This is perhaps best
illustrated by the classical k-center problem: there is a simple tight
2-approximation algorithm for the uncapacitated version whereas the first
constant factor approximation algorithm for the general version with capacities
was only recently obtained by using an intricate rounding algorithm that
achieves an approximation guarantee in the hundreds.
Our paper aims to bridge this discrepancy. For the capacitated k-center
problem, we give a simple algorithm with a clean analysis that allows us to
prove an approximation guarantee of 9. It uses the standard LP relaxation and
comes close to settling the integrality gap (after necessary preprocessing),
which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first
reducing to special tree instances, and then solves such instances optimally.
Our concept of tree instances is quite versatile, and applies to natural
variants of the capacitated k-center problem for which we also obtain improved
algorithms. Finally, we give evidence to show that more powerful preprocessing
could lead to better algorithms, by giving an approximation algorithm that
beats the integrality gap for instances where all non-zero capacities are
uniform.Comment: 21 pages, 2 figure
LP-Based Algorithms for Capacitated Facility Location
Linear programming has played a key role in the study of algorithms for
combinatorial optimization problems. In the field of approximation algorithms,
this is well illustrated by the uncapacitated facility location problem. A
variety of algorithmic methodologies, such as LP-rounding and primal-dual
method, have been applied to and evolved from algorithms for this problem.
Unfortunately, this collection of powerful algorithmic techniques had not yet
been applicable to the more general capacitated facility location problem. In
fact, all of the known algorithms with good performance guarantees were based
on a single technique, local search, and no linear programming relaxation was
known to efficiently approximate the problem.
In this paper, we present a linear programming relaxation with constant
integrality gap for capacitated facility location. We demonstrate that the
fundamental theories of multi-commodity flows and matchings provide key
insights that lead to the strong relaxation. Our algorithmic proof of
integrality gap is obtained by finally accessing the rich toolbox of LP-based
methodologies: we present a constant factor approximation algorithm based on
LP-rounding.Comment: 25 pages, 6 figures; minor revision
Optimal Design of Wireless Mesh Networks
Wireless Mesh Networks (WMNs) are cost-effective and provide an appealing answer to connectivity issues of ubiquitous computing. Unfortunately, wireless networks are known for strong waste of capacity when their size in- creases. Thus, a key challenge for network operators is to provide guaranteed quality of service. Maximizing network capacity requires to optimize jointly the gateways placement, the routing and the link scheduling taking interferences into account. We present MILP models for computing an optimal 802.11a or 802.16 WMN design providing max-min bandwidth guarantee
Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location
Metric facility location is a well-studied problem for which linear
programming methods have been used with great success in deriving approximation
algorithms. The capacity-constrained generalizations, such as capacitated
facility location (CFL) and lower-bounded facility location (LBFL), have proved
notorious as far as LP-based approximation is concerned: while there are
local-search-based constant-factor approximations, there is no known linear
relaxation with constant integrality gap. According to Williamson and Shmoys
devising a relaxation-based approximation for \cfl\ is among the top 10 open
problems in approximation algorithms.
This paper advances significantly the state-of-the-art on the effectiveness
of linear programming for capacity-constrained facility location through a host
of impossibility results for both CFL and LBFL. We show that the relaxations
obtained from the natural LP at levels of the Sherali-Adams
hierarchy have an unbounded gap, partially answering an open question of
\cite{LiS13, AnBS13}. Here, denotes the number of facilities in the
instance. Building on the ideas for this result, we prove that the standard CFL
relaxation enriched with the generalized flow-cover valid inequalities
\cite{AardalPW95} has also an unbounded gap. This disproves a long-standing
conjecture of \cite{LeviSS12}. We finally introduce the family of proper
relaxations which generalizes to its logical extreme the classic star
relaxation and captures general configuration-style LPs. We characterize the
behavior of proper relaxations for CFL and LBFL through a sharp threshold
phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599
Online Mixed Packing and Covering
Recent work has shown that the classical framework of
solving optimization problems by obtaining a fractional
solution to a linear program (LP) and rounding it to
an integer solution can be extended to the online setting
using primal-dual techniques. The success of this
new framework for online optimization can be gauged
from the fact that it has led to progress in several longstanding open questions. However, to the best of our
knowledge, this framework has previously been applied
to LPs containing only packing or only covering constraints,
or minor variants of these. We extend this
framework in a fundamental way by demonstrating that
it can be used to solve mixed packing and covering LPs
online, where packing constraints are given offline and
covering constraints are received online. The objective
is to minimize the maximum multiplicative factor by
which any packing constraint is violated, while satisfying
the covering constraints. Our results represent the
first algorithm that obtains a polylogarithmic competitive
ratio for solving mixed LPs online.
We then consider two canonical examples of mixed
LPs: unrelated machine scheduling with startup costs,
and capacity constrained facility location. We use ideas
generated from our result for mixed packing and covering
to obtain polylogarithmic-competitive algorithms
for these problems. We also give lower bounds to show
that the competitive ratios of our algorithms are nearly
tight