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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.34

Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

Authors: Chao Liao, Jiabao Lin, Pinyan Lu, and Zhenyu Mao

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph.

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Chao Liao, Jiabao Lin, Pinyan Lu, and Zhenyu Mao. Counting Independent Sets and Colorings on Random Regular Bipartite Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{liao_et_al:LIPIcs.APPROX-RANDOM.2019.34,
  author =	{Liao, Chao and Lin, Jiabao and Lu, Pinyan and Mao, Zhenyu},
  title =	{{Counting Independent Sets and Colorings on Random Regular Bipartite Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{34:1--34:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.34},
  URN =		{urn:nbn:de:0030-drops-112498},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.34},
  annote =	{Keywords: Approximate counting, Polymer model, Hardcore model, Coloring, Random bipartite graphs}
}

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Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

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