# Approximation Algorithms for Graph Homomorphism Problems

• Michael Langberg
• Yuval Rabani
• Chaitanya Swamy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4110)

## 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 $${\ensuremath{\varphi}}':U\mapsto V_H,\ U{\subseteq} V_G$$, 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 $$\frac{6}{7}\simeq 0.8571$$, showing that our guarantee is almost tight. For maximum graph homomorphism, we show that a $$\bigl({\ensuremath{\frac{1}{2}}}+{\ensuremath{\varepsilon}}_0)$$-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 $$\bigl({\ensuremath{\frac{1}{2}}}+\Omega(\frac{1}{|H|\log{|H|}})\bigr)$$-approximation algorithm.

## Keywords

Approximation Algorithm Random Graph Linear Programming Relaxation Label Graph Graph Homomorphism
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Michael Langberg
• 1
• Yuval Rabani
• 2
• Chaitanya Swamy
• 3