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Approximation Algorithms for Graph Homomorphism Problems

Langberg, Michael and Rabani, Yuval and Swamy, Chaitanya (2006) Approximation Algorithms for Graph Homomorphism Problems. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Lecture Notes in Computer Science. No.4110. Springer , Berlin, pp. 176-187. ISBN 978-3-540-38044-3. https://resolver.caltech.edu/CaltechAUTHORS:20191112-105626756

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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 φ′:U↦V_H, 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+ε_0)-approximation algorithm, for any constant ε_0 > 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.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/11830924_18DOIArticle
https://rdcu.be/b3YlHPublisherFree ReadCube access
ORCID:
AuthorORCID
Langberg, Michael0000-0002-7470-0718
Additional Information:© 2006 Springer-Verlag Berlin Heidelberg. Research supported in part by NSF grant CCF-0346991. Supported in part by ISF 52/03, BSF 2002282, and the Fund for the Promotion of Research at the Technion. Part of this work was done while visiting Caltech.
Funders:
Funding AgencyGrant Number
NSFCCF-0346991
Israel Science Foundation52/03
Binational Science Foundation (USA-Israel)2002282
TechnionUNSPECIFIED
Subject Keywords:Approximation Algorithm; Random Graph; Linear Programming Relaxation; Label Graph; Graph Homomorphism
Series Name:Lecture Notes in Computer Science
Issue or Number:4110
Record Number:CaltechAUTHORS:20191112-105626756
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20191112-105626756
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:99807
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Nov 2019 22:55
Last Modified:05 May 2020 21:10

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