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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2019.91

Toward a Dichotomy for Approximation of H-Coloring

Authors: Akbar Rafiey, Arash Rafiey, and Thiago Santos

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract

Given two (di)graphs G, H and a cost function c:V(G) x V(H) -> Q_{>= 0} cup {+infty}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f:V(G)-> V(H) (a.k.a H-coloring) that minimizes sum limits_{v in V(G)}c(v,f(v)). The complexity of exact minimization of this problem is well understood [Pavol Hell and Arash Rafiey, 2012], 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. In terms of 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 (i.e. symmetric digraph). 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: - 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; - MinHOM(H) has a |V(H)|^2-approximation algorithm if H is a bi-arc digraph; - 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 |V(H)|.

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Akbar Rafiey, Arash Rafiey, and Thiago Santos. Toward a Dichotomy for Approximation of H-Coloring. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 91:1-91:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{rafiey_et_al:LIPIcs.ICALP.2019.91,
  author =	{Rafiey, Akbar and Rafiey, Arash and Santos, Thiago},
  title =	{{Toward a Dichotomy for Approximation of H-Coloring}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{91:1--91:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.91},
  URN =		{urn:nbn:de:0030-drops-106678},
  doi =		{10.4230/LIPIcs.ICALP.2019.91},
  annote =	{Keywords: Approximation algorithms, minimum cost homomorphism, randomized rounding}
}

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Toward a Dichotomy for Approximation of H-Coloring

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