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 Hcoloring) 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 biarc digraphs (digraphs with a conservative semilattice polymorphism or minordering), and karc digraphs (digraphs with an extended minordering). Specifically, we show that:
 Dichotomy for Graphs: MinHOM(H) has a 2V(H)approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a biarc graph), otherwise, it is inapproximable;
 MinHOM(H) has a V(H)^2approximation algorithm if H is a biarc digraph;
 MinHOM(H) has a V(H)^2approximation algorithm if H is a karc 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 HColoring}},
booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages = {91:191:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771092},
ISSN = {18688969},
year = {2019},
volume = {132},
editor = {Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10667},
URN = {urn:nbn:de:0030drops106678},
doi = {10.4230/LIPIcs.ICALP.2019.91},
annote = {Keywords: Approximation algorithms, minimum cost homomorphism, randomized rounding}
}
Keywords: 

Approximation algorithms, minimum cost homomorphism, randomized rounding 
Collection: 

46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) 
Issue Date: 

2019 
Date of publication: 

04.07.2019 