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We determine the computational complexity of approximately counting and sampling independent sets of a given size in boundeddegree graphs. That is, we identify a critical density αc(∆) and provide (i) for α < αc(∆) randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most αn in n-vertex graphs of maximum degree ∆; and (ii) a proof that unless NP=RP, no such algorithms exist for α > αc(∆). The critical density is the occupancy fraction of hard core model on the clique K∆+1 at the uniqueness threshold on the infinite ∆-regular tree, giving αc(∆) ∼ e 1+e 1 ∆ as ∆ → ∞.

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