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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.91
URN: urn:nbn:de:0030-drops-106678
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10667/
Rafiey, Akbar ;
Rafiey, Arash ;
Santos, Thiago
Toward a Dichotomy for Approximation of H-Coloring
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)|.
BibTeX - Entry
@InProceedings{rafiey_et_al:LIPIcs:2019:10667,
author = {Akbar Rafiey and Arash Rafiey and Thiago Santos},
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 = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10667},
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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Keywords: |
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Approximation algorithms, minimum cost homomorphism, randomized rounding |
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Collection: |
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46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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04.07.2019 |
