Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs
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 \(\alpha_c(\Delta )\) and provide (i) for \(\alpha \lt \alpha_c(\Delta )\) 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 \(\alpha \gt \alpha_c(\Delta )\) . The critical density is the occupancy fraction of the hard-core model on the complete graph \(K_{\Delta+1}\) at the uniqueness threshold on the infinite \(\Delta\) -regular tree, giving \(\alpha_c(\Delta )\sim \frac{e}{1+e}\frac{1}{\Delta }\) as \(\Delta \to \infty\) . Our methods apply more generally to antiferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.
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© 2023 Society for Industrial and Applied Mathematics.
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Submitted: 17 December 2021
Accepted: 23 January 2023
Published online: 26 April 2023
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National Science Foundation (NSF): DMS-1847451, CCF-1934915
Funding: The second author was supported in part by NSF grants DMS-1847451 and CCF-1934915.
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