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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.297
URN: urn:nbn:de:0030-drops-47046
URL: https://drops.dagstuhl.de/opus/volltexte/2014/4704/
Kolliopoulos, Stavros G. ;
Moysoglou, Yannis
Sherali-Adams Gaps, Flow-cover Inequalities and Generalized Configurations for Capacity-constrained Facility Location
Abstract
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 Omega(n) levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question from the literature. Here, n 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 has also an unbounded gap. This disproves a long-standing conjecture of Levi et al. 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.
BibTeX - Entry
@InProceedings{kolliopoulos_et_al:LIPIcs:2014:4704,
author = {Stavros G. Kolliopoulos and Yannis Moysoglou},
title = {{Sherali-Adams Gaps, Flow-cover Inequalities and Generalized Configurations for Capacity-constrained Facility Location}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
pages = {297--312},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-74-3},
ISSN = {1868-8969},
year = {2014},
volume = {28},
editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4704},
URN = {urn:nbn:de:0030-drops-47046},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.297},
annote = {Keywords: Approximation Algorithms, Linear Programming, Facility Location}
}
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Keywords: |
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Approximation Algorithms, Linear Programming, Facility Location |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014) |
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Issue Date: |
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2014 |
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Date of publication: |
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04.09.2014 |
