Approximation algorithms for graph homomorphism problems

Michael Langberg*, Yuval Rabani, Chaitanya Swamy

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

22 Scopus citations

Abstract

We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping φ : VG → VH that maximizes the number of edges of G that are mapped to edges of H. This problem encodes various fundamental NP-hardproblems 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 ⊆ VG. We want to partition V G into |T| parts, each containing exactly one terminal, so as to maximize the number of edges in EG 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 φ′ : U → VH, U ⊆ 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 6/7 ≃ 0.8571, showing that our guarantee is almost tight. For maximum graph homomorphism, we show that a (1/2 + ε0)-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 (1/2 + Ω(1/|H|log|H|))-approximation algorithm.

Original languageAmerican English
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a
PublisherSpringer Verlag
Pages176-187
Number of pages12
ISBN (Print)3540380442, 9783540380443
DOIs
StatePublished - 2006
Externally publishedYes
Event9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 - Barcelona, Spain
Duration: 28 Aug 200630 Aug 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4110 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006
Country/TerritorySpain
CityBarcelona
Period28/08/0630/08/06

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