License
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.46
URN: urn:nbn:de:0030-drops-63262
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6326/
Go to the corresponding LIPIcs Volume Portal


Galanis, Andreas ; Goldberg, Leslie Ann ; Jerrum, Mark

A Complexity Trichotomy for Approximately Counting List H-Colourings

pdf-format:
LIPIcs-ICALP-2016-46.pdf (0.5 MB)


Abstract

We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem which is believed to be of intermediate complexity - it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colourings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (i) it has a natural formulation in terms of hereditary graph classes, and (ii) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.

BibTeX - Entry

@InProceedings{galanis_et_al:LIPIcs:2016:6326,
  author =	{Andreas Galanis and Leslie Ann Goldberg and Mark Jerrum},
  title =	{{A Complexity Trichotomy for Approximately Counting List H-Colourings}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{46:1--46:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6326},
  URN =		{urn:nbn:de:0030-drops-63262},
  doi =		{10.4230/LIPIcs.ICALP.2016.46},
  annote =	{Keywords: approximate counting, graph homomorphisms, list colourings}
}

Keywords: approximate counting, graph homomorphisms, list colourings
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016


DROPS-Home | Imprint | Privacy Published by LZI