Approximation algorithms for graph homomorphism problems

• Authors: Michael Langberg;Yuval Rabani;Chaitanya Swamy
• Venue: APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
• Year: 2006

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 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 ${\ensuremath{\varphi}}':U\mapsto V_H,\ U{\subseteq} V_G$, 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 ε00, 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.