Inapproximability of the independent set polynomial in the complex plane
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We study the complexity of approximating the independent set polynomial Z_G(λ) of a graph G with maximum degree Δ when the activity λ is a complex number. This problem is already well understood when λ is real using connections to the Δ-regular tree T. The key concept in that case is the "occupation ratio" of the tree T. This ratio is the contribution to Z_T(λ) from independent sets containing the root of the tree, divided by Z_T(λ) itself. If λ is such that the occupation ratio converges to a limit, as the height of T grows, then there is an FPTAS for approximating Z_G(λ) on a graph G with maximum degree Δ. Otherwise, the approximation problem is NP-hard. Unsurprisingly, the case where λ is complex is more challenging. Peters and Regts identified the values of λ for which the occupation ratio of the Δ-regular tree converges. These values carve a cardioid-shaped region Λ_Δ in the complex plane. Motivated by the picture in the real case, they asked whether Λ_Δ marks the true approximability threshold for general complex values λ. Our main result shows that for every λ outside of Λ_Δ, the problem of approximating Z_G(λ) on graphs G with maximum degree at most Δ is indeed NP-hard. In fact, when λ is outside of Λ_Δ and is not a positive real number, we give the stronger result that approximating Z_G(λ) is actually #P-hard. If λ is a negative real number outside of Λ_Δ, we show that it is #P-hard to even decide whether Z_G(λ)>0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis - specifically the study of iterative multivariate rational maps.
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