Approximation Algorithms for Graph Homomorphism Problems
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.
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.Additional details
- Eprint ID
- 99807
- DOI
- 10.1007/11830924_18
- Resolver ID
- CaltechAUTHORS:20191112-105626756
- CCF-0346991
- NSF
- 52/03
- Israel Science Foundation
- 2002282
- Binational Science Foundation (USA-Israel)
- Technion
- Created
-
2019-11-12Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field
- Series Name
- Lecture Notes in Computer Science
- Series Volume or Issue Number
- 4110