Wrochna, Marcin
Homomorphism Reconfiguration via Homotopy
Abstract
We consider the following problem for a fixed graph H: given a graph G and two Hcolorings of G, i.e. homomorphisms from G to H, can one be transformed into the other by changing one color at a time, maintaining an Hcoloring throughout.This is the same as finding a path in the Hom(G,H) complex. For H=K_k this is the problem of finding paths between kcolorings, which was recently shown to be in P for k\leq 3 and PSPACEcomplete otherwise (Bonsma and Cereceda 2009, Cereceda et al. 2011).
We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H with no C_4 subgraph. This gives a large class of constraints for which finding solutions to the Constraint Satisfaction Problem is NPcomplete, but paths in the solution space can be found in polynomial time.
The algorithm uses a characterization of possible reconfiguration sequences (that is, paths in Hom(G,H)), whose main part is a purely topological condition described in terms of the fundamental groupoid of H seen as a topological space.
BibTeX  Entry
@InProceedings{wrochna:LIPIcs:2015:4954,
author = {Marcin Wrochna},
title = {{Homomorphism Reconfiguration via Homotopy}},
booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
pages = {730742},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897781},
ISSN = {18688969},
year = {2015},
volume = {30},
editor = {Ernst W. Mayr and Nicolas Ollinger},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/4954},
URN = {urn:nbn:de:0030drops49546},
doi = {10.4230/LIPIcs.STACS.2015.730},
annote = {Keywords: reconfiguration, recoloring, homomorphisms, homotopy, hom complex}
}
2015
Keywords: 

reconfiguration, recoloring, homomorphisms, homotopy, hom complex 
Seminar: 

32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Issue date: 

2015 
Date of publication: 

2015 