Toward a Dichotomy for Approximation of H-coloring [article]

Akbar Rafiey, Arash Rafiey, Thiago Santos
2019 arXiv   pre-print
Given two (di)graphs G, H and a cost function c:V(G)× V(H) →Q_≥ 0∪{+∞}, in the minimum cost homomorphism problem, MinHOM(H), goal is finding a homomorphism f:V(G)→ V(H) (a.k.a H-coloring) that minimizes ∑_v∈ V(G)c(v,f(v)). The complexity of exact minimization of this problem is well understood [34], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs. In this paper, we consider the approximation of MinHOM within a constant factor.
more » ... r digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph. For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that: 1. Dichotomy for Graphs: MinHOM(H) has a 2|V(H)|-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable; 2. MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a bi-arc digraph; 3. MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a k-arc digraph. In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of H.
arXiv:1902.02201v3 fatcat:hvl3idprorehvhyrt5mgydbgni