Approximation Algorithms for Graph Homomorphism Problems

We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping φ : VG 7→ 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 graphG 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 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 7→ 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 ' 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.