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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
FPT Approximation for Constrained Metric k-Median/Means
The Metric -median problem over a metric space is
defined as follows: given a set of facility locations
and a set of clients, open a set of
facilities such that the total service cost, defined as , is minimised. The metric -means
problem is defined similarly using squared distances. In many applications
there are additional constraints that any solution needs to satisfy. This gives
rise to different constrained versions of the problem such as -gather,
fault-tolerant, outlier -means/-median problem. Surprisingly, for many of
these constrained problems, no constant-approximation algorithm is known. We
give FPT algorithms with constant approximation guarantee for a range of
constrained -median/means problems. For some of the constrained problems,
ours is the first constant factor approximation algorithm whereas for others,
we improve or match the approximation guarantee of previous works. We work
within the unified framework of Ding and Xu that allows us to simultaneously
obtain algorithms for a range of constrained problems. In particular, we obtain
a -approximation and -approximation for the
constrained versions of the -median and -means problem respectively in
FPT time. In many practical settings of the -median/means problem, one is
allowed to open a facility at any client location, i.e., . For
this special case, our algorithm gives a -approximation and
-approximation for the constrained versions of -median and
-means problem respectively in FPT time. Since our algorithm is based on
simple sampling technique, it can also be converted to a constant-pass
log-space streaming algorithm
Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem
Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under various settings reflecting typical structures within the optimization problems of practical interest, little is known on how the problem behaves in conjunction with parity constraints. This shortfall of understanding was rather discouraging when we consider the central role of parity in the field of combinatorics.
In this paper, we present the first constant-factor approximation algorithm for the facility location problem with parity constraints. We are given as the input a metric on a set of facilities and clients, the opening cost of each facility, and the parity requirement - odd, even, or unconstrained - of every facility in this problem. The objective is to open a subset of facilities and assign every client to an open facility so as to minimize the sum of the total opening costs and the assignment distances, but subject to the condition that the number of clients assigned to each open facility must have the same parity as its requirement.
Although the unconstrained facility location problem as a relaxation for this parity-constrained generalization has unbounded gap, we demonstrate that it yields a structured solution whose parity violation can be corrected at small cost. This correction is prescribed by a T-join on an auxiliary graph constructed by the algorithm. This auxiliary graph does not satisfy the triangle inequality, but we show that a carefully chosen set of shortcutting operations leads to a cheap and sparse T-join. Finally, we bound the correction cost by exhibiting a combinatorial multi-step construction of an upper bound
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