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Register a new userApproximately counting independent sets of a given size in bounded-degree graphs
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Added by
Will Perkins
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 $alpha_c(Delta)$ and provide (i) for $alpha < alpha_c(Delta)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $alpha n$ in $n$-vertex graphs of maximum degree $Delta$; and (ii) a proof that unless NP=RP, no such algorithms exist for $alpha>alpha_c(Delta)$. The critical density is the occupancy fraction of hard core model on the clique $K_{Delta+1}$ at the uniqueness threshold on the infinite $Delta$-regular tree, giving $alpha_c(Delta)simfrac{e}{1+e}frac{1}{Delta}$ as $Deltatoinfty$.
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We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.
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