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 and provide (i) for randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most in -vertex graphs of maximum degree , and (ii) a proof that unless NP = RP, no such algorithms exist for . The critical density is the occupancy fraction of the hard-core model on the complete graph at the uniqueness threshold on the infinite -regular tree, giving as . Our methods apply more generally to antiferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.