### Approximation Algorithms for Graph Homomorphism Problems [chapter]

Michael Langberg, Yuval Rabani, Chaitanya Swamy
2006 Lecture Notes in Computer Science
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-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 ⊆ VG. We want to partition VG into |T | parts, each containing exactly one terminal, so as to maximize the number of
more » ... s 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 ⊆ VG, 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 1 Introduction We introduce the maximum graph homomorphism (MGH) problem: given a graph G = (V G , E G ) and a target or "label" graph H = (V H , E 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 is trivially NP-hard; for example, deciding if G is k-colorable is equivalent to checking if the solution to MGH with graph G and the target graph H being a k-clique, has value |E G |. Several fundamental NP-hard optimization problems can be encoded easily as special cases of MGH.