Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs

Authors Ewan Davies , Will Perkins

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Author Details

Ewan Davies
  • Department of Computer Science, University of Colorado, Boulder, CO, USA
Will Perkins
  • Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, IL, USA

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Ewan Davies and Will Perkins. Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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 → ∞.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Random walks and Markov chains
  • approximate counting
  • independent sets
  • Markov chains


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