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URN: urn:nbn:de:0030-drops-141310
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Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs

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Abstract

We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density α_c(Δ) and provide (i) for α < α_c(Δ) randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most α n in n-vertex graphs of maximum degree Δ; and (ii) a proof that unless NP=RP, no such algorithms exist for α > α_c(Δ). The critical density is the occupancy fraction of hard core model on the clique K_{Δ+1} at the uniqueness threshold on the infinite Δ-regular tree, giving α_c(Δ) ~ e/(1+e)1/(Δ) as Δ → ∞.

BibTeX - Entry

@InProceedings{davies_et_al:LIPIcs.ICALP.2021.62,
  author =	{Davies, Ewan and Perkins, Will},
  title =	{{Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{62:1--62:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14131},
  URN =		{urn:nbn:de:0030-drops-141310},
  doi =		{10.4230/LIPIcs.ICALP.2021.62},
  annote =	{Keywords: approximate counting, independent sets, Markov chains}
}

Keywords: approximate counting, independent sets, Markov chains
Seminar: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Issue date: 2021
Date of publication: 02.07.2021


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