Article

Approximation algorithms for graph homomorphism problems

Authors:

Michael Langberg,

Chaitanya SwamyAuthors Info & Claims

APPROX'06/RANDOM'06: Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation

Pages 176 - 187

https://doi.org/10.1007/11830924_18

Published: 28 August 2006 Publication History

Abstract

We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping ϕ:V_G↦V_H that maximizes the number of edges of G that are mapped to edges of H. This problem encodes various fundamental NP-hard problems including Maxcut and Max-k-cut. We also consider the multiway uncut problem. We are given a graph G and a set of terminals T⊆V_G. We want to partition V_G into |T| parts, each containing exactly one terminal, so as to maximize the number of edges in E_G having both endpoints in the same part. Multiway uncut can be viewed as a special case of prelabeled MGH where one is also given a prelabeling ${\ensuremath{\varphi}}':U\mapsto V_H,\ U{\subseteq} V_G$, and the output has to be an extension of ϕ′.

Both MGH and multiway uncut have a trivial 0.5-approximation algorithm. We present a 0.8535-approximation algorithm for multiway uncut based on a natural linear programming relaxation. This relaxation has an integrality gap of $\frac{6}{7}\simeq 0.8571$, showing that our guarantee is almost tight. For maximum graph homomorphism, we show that a $\bigl({\ensuremath{\frac{1}{2}}}+{\ensuremath{\varepsilon}}_0)$-approximation algorithm, for any constant ε₀>0, implies an algorithm for distinguishing between certain average-case instances of the subgraph isomorphism problem that appear to be hard. Complementing this, we give a $\bigl({\ensuremath{\frac{1}{2}}}+\Omega(\frac{1}{|H|\log{|H|}})\bigr)$-approximation algorithm.

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Romero MWrochna MŽivný SMarx D(2021)Treewidth-pliability and PTAS for max-CSPsProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458093(473-483)Online publication date: 10-Jan-2021
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Approximation algorithms for graph homomorphism problems
1. Mathematics of computing
  1. Discrete mathematics
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cover image Guide Proceedings

APPROX'06/RANDOM'06: Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation

August 2006

521 pages

ISBN:3540380442

Editors:
Josep Díaz
Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord - Ed. Omega, 240 Jordi Girona Salgado, Barcelona
,
Klaus Jansen
Institute for Computer Science, University of Kiel, Olshausenstrasse 40, Kiel, Germany
,
José P. Rolim
Institute for Computer Science, Centre Universitaire d'Informatique, Battelle Bâtiment A, Route de Drize 7,, Carouge, Geneva, Switzerland
,
Uri Zwick
School of Computer Science, Tel Aviv University, Battelle Bâtiment A, Route de Drize 7,, Tel Aviv, Geneva, Israel

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Berlin, Heidelberg

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Published: 28 August 2006

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Cited By

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Romero MWrochna MŽivný SMarx D(2021)Treewidth-pliability and PTAS for max-CSPsProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458093(473-483)Online publication date: 10-Jan-2021
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https://dl.acm.org/doi/10.14778/3407790.3407834
Zhang PLiu Z(2020)Approximating Max k-Uncut via LP-rounding Plus Greed, with Applications to Densest k-SubgraphAlgorithmic Aspects in Information and Management10.1007/978-3-030-57602-8_15(161-172)Online publication date: 10-Aug-2020
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Zhang PXu YJiang TLi ALin GMiyano E(2018)Improved Approximation Algorithms for the Maximum Happy Vertices and Edges ProblemsAlgorithmica10.1007/s00453-017-0302-880:5(1412-1438)Online publication date: 1-May-2018
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Arvind VKöbler JKuhnert SVasudev Y(2012)Approximate graph isomorphismProceedings of the 37th international conference on Mathematical Foundations of Computer Science10.1007/978-3-642-32589-2_12(100-111)Online publication date: 27-Aug-2012
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