Implementations and the independent set polynomial below the Shearer threshold

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ≥2, the Shearer threshold value λ⁎(Δ)=(Δ−1)Δ−1/ΔΔ. It known that for λ<−λ⁎(Δ), there are graphs G with maximum degree Δ whose polynomial, evaluated at λ, most 0. Also, no such any λ>−λ⁎(Δ). This paper motivated by computational problem approximating when λ<−λ⁎(Δ). key issue complexity bounds this “implementation”. Informally, an implementation a real number λ′ graph hard-core partition function, simulates vertex-weight sense ratio between contribution to function from sets containing certain vertex do not contain vertex. Implementations cornerstone intractability results approximately evaluating polynomial. Our main result that, it possible implement values dense over reals. tight reals λ>λ⁎(Δ). has already been used Bezáková (STOC 2018) show #P-hard approximate evaluation on In appendix, we give additional incomparable inapproximability (strengthening bound exponential factor, but weakening hardness NP-hardness).