Curticapean, Radu ;
Dell, Holger ;
Roth, Marc
Counting EdgeInjective Homomorphisms and Matchings on Restricted Graph Classes
Abstract
We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W[1]hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)poly(n) time algorithms. We show that #W[1]hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side.
To prove the result for line graphs, we observe that kmatchings in line graphs can be equivalently viewed as edgeinjective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs. Here, a homomorphism from H to G is edgeinjective if it maps any two distinct edges of H to distinct edges in G. We show that edgeinjective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertexcover number after removing isolated edges. For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edgeinjective homomorphisms, restricted to patterns from H, is #W[1]hard if no such bound exists.
Our proofs rely on an edgecolored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.
BibTeX  Entry
@InProceedings{curticapean_et_al:LIPIcs:2017:7008,
author = {Radu Curticapean and Holger Dell and Marc Roth},
title = {{Counting EdgeInjective Homomorphisms and Matchings on Restricted Graph Classes}},
booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
pages = {25:125:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770286},
ISSN = {18688969},
year = {2017},
volume = {66},
editor = {Heribert Vollmer and Brigitte Vallée},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7008},
URN = {urn:nbn:de:0030drops70080},
doi = {10.4230/LIPIcs.STACS.2017.25},
annote = {Keywords: matchings, homomorphisms, line graphs, counting complexity, parameterized complexity}
}
2017
Keywords: 

matchings, homomorphisms, line graphs, counting complexity, parameterized complexity 
Seminar: 

34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Issue date: 

2017 
Date of publication: 

2017 