Computing the Independence Polynomial: from the Tree Threshold down to the Roots
We study an algorithm for approximating the multivariate independence polynomial $Z(\mathbf{z})$, with negative and complex arguments, an object that has strong connections to combinatorics and to statistical physics. In particular, the independence polynomial with negative arguments, $Z(-\mathbf{p})$, determines the Shearer region, the maximal region of probabilities to which the Lovasz Local Lemma (LLL) can be extended (Shearer 1985). In statistical physics, complex zeros of the independence polynomial relate to existence of phase transitions. Our main result is a deterministic algorithm to compute approximately the independence polynomial in any root-free complex polydisc centered at the origin. Our algorithm is essentially the same as Weitz's algorithm for positive parameters up to the tree uniqueness threshold, and the core of our analysis is a novel multivariate form of the correlation decay technique, which can handle non-uniform complex parameters. In particular, in the univariate real setting our work implies that Weitz's algorithm works in an interval between two critical points $(λ'_c(d), λ_c(d))$, and outside of this interval an approximation of $Z(\mathbf{z})$ is known to be NP-hard. As an application, we give a sub-exponential time algorithm for testing approximate membership in the Shearer region. We also give a new rounding based deterministic algorithm for Shearer's lemma (an extension of the LLL), which, however, runs in sub-exponential time. On the hardness side, we prove that evaluating $Z(\mathbf{z})$ at an arbitrary point in Shearer's region, and testing membership in Shearer's region, are #P-hard problems. We also establish the best possible dependence of the exponent of the run time of Weitz's correlation decay technique in the negative regime on the distance to the boundary of the Shearer region.