Integrality gaps for colorful matchings

Steven Kelk; Georgios Stamoulis

doi:10.1016/j.disopt.2018.12.003

Integrality gaps for colorful matchings

Steven Kelk, Georgios Stamoulis^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

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Abstract

We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Furedi (1981). We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem. (C) 2018 Elsevier B.V. All rights reserved.

Original language	English
Pages (from-to)	73-92
Number of pages	20
Journal	Discrete Optimization
Volume	32
Early online date	12 Jan 2019
DOIs	https://doi.org/10.1016/j.disopt.2018.12.003
Publication status	Published - May 2019

Keywords

APPROXIMATION ALGORITHMS
COMPLEXITY
Colorful matchings
Integrality gap
LIFT
Lift and project
Linear programming
RELAXATIONS
SEMIDEFINITE
SET
SHERALI-ADAMS
Sherali-Adams
TRANSVERSAL
WEIGHT

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10.1016/j.disopt.2018.12.003

Full TextFinal published version, 938 KBLicence: Taverne

https://linkinghub.elsevier.com/retrieve/pii/S1572528618300239

Cite this

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title = "Integrality gaps for colorful matchings",

abstract = "We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams {"}lift-and-project{"} technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Furedi (1981). We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem. (C) 2018 Elsevier B.V. All rights reserved.",

keywords = "APPROXIMATION ALGORITHMS, COMPLEXITY, Colorful matchings, Integrality gap, LIFT, Lift and project, Linear programming, RELAXATIONS, SEMIDEFINITE, SET, SHERALI-ADAMS, Sherali-Adams, TRANSVERSAL, WEIGHT",

author = "Steven Kelk and Georgios Stamoulis",

note = "Funding Information: The authors would like to sincerely thank the two anonymous referees and the editorial for carefully reading a first version of theirmanuscript and for the many helpful comments and suggestions that greatly improved its content and presentation. The second author acknowledges the support of an NWO TOP 2 grant ( 617.001.301 ). Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2019",

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doi = "10.1016/j.disopt.2018.12.003",

language = "English",

volume = "32",

pages = "73--92",

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T1 - Integrality gaps for colorful matchings

AU - Kelk, Steven

AU - Stamoulis, Georgios

N1 - Funding Information: The authors would like to sincerely thank the two anonymous referees and the editorial for carefully reading a first version of theirmanuscript and for the many helpful comments and suggestions that greatly improved its content and presentation. The second author acknowledges the support of an NWO TOP 2 grant ( 617.001.301 ). Publisher Copyright: © 2018 Elsevier B.V.

PY - 2019/5

Y1 - 2019/5

N2 - We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Furedi (1981). We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem. (C) 2018 Elsevier B.V. All rights reserved.

AB - We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Furedi (1981). We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem. (C) 2018 Elsevier B.V. All rights reserved.

KW - APPROXIMATION ALGORITHMS

KW - COMPLEXITY

KW - Colorful matchings

KW - Integrality gap

KW - LIFT

KW - Lift and project

KW - Linear programming

KW - RELAXATIONS

KW - SEMIDEFINITE

KW - SET

KW - SHERALI-ADAMS

KW - Sherali-Adams

KW - TRANSVERSAL

KW - WEIGHT

U2 - 10.1016/j.disopt.2018.12.003

DO - 10.1016/j.disopt.2018.12.003

M3 - Article

SN - 1572-5286

VL - 32

SP - 73

EP - 92

JO - Discrete Optimization

JF - Discrete Optimization

ER -