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.
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Langberg, M., Rabani, Y., Swamy, C. (2006). Approximation Algorithms for Graph Homomorphism Problems. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2006 2006. Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11830924_18
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DOI: https://doi.org/10.1007/11830924_18
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