Approximately counting independent sets of a given size in bounded-degree graphs

TL;DR

This paper characterizes the computational complexity of approximately counting and sampling independent sets of a specified size in bounded-degree graphs, identifying a critical density threshold that determines algorithmic feasibility.

Contribution

It introduces a precise threshold for the existence of efficient algorithms for counting and sampling independent sets of a given size in bounded-degree graphs, linking it to the occupancy fraction of the hard core model.

Findings

Efficient algorithms exist for densities below the critical threshold.

No such algorithms are possible above the threshold unless NP=RP.

The critical density relates to the occupancy fraction at the uniqueness threshold on the infinite regular tree.

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 $\alpha < \alpha_c(\Delta)$ 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 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 anti-ferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.