TY - GEN
T1 - Approximation algorithms for graph homomorphism problems
AU - Langberg, Michael
AU - Rabani, Yuval
AU - Swamy, Chaitanya
PY - 2006
Y1 - 2006
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=33750074414&partnerID=8YFLogxK
U2 - 10.1007/11830924_18
DO - 10.1007/11830924_18
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AN - SCOPUS:33750074414
SN - 3540380442
SN - 9783540380443
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 176
EP - 187
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a
PB - Springer Verlag
T2 - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006
Y2 - 28 August 2006 through 30 August 2006
ER -