2026-08-16T22:57:54Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:14131 2024-03-06T10:53:33Z ddc:004 open_access

Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs Davies, Ewan Perkins, Will approximate counting independent sets Markov chains 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 → ∞. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ewan Davies and Will Perkins 2021 Is Part Of LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2021.62 urn:nbn:de:0030-drops-141310 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.62 eng https://creativecommons.org/licenses/by/4.0/legalcode