Computing the Independence Polynomial: from the Tree Threshold down to the Roots

The independence polynomial has been widely studied in algebraic graph theory, in statistical physics, and in algorithms for counting and sampling problems. Seminal results of Weitz (2006) and Sly (2010) have shown that in bounded-degree graphs the independence polynomial can be eXciently approximated if the argument is positive and below a certain threshold, whereas above that threshold the polynomial is hard to approximate. Furthermore, this threshold exactly corresponds to a phase transition in physics, which demarcates the region within which the Gibbs measure has correlation decay. Evaluating the independence polynomial with negative or complex arguments may not have a counting interpretation, but it does have strong connections to combinatorics and to statistical physics. The independence polynomial with negative arguments determines the maximal region of probabilities to which the Lovász Local Lemma (LLL) can be extended, and also gives a lower bound on the probability in the LLL’s conclusion (Shearer 1985). In statistical physics, complex zeros of the independence polynomial relate to existence of phase transitions, and there is a relation between negative and complex zeros (Penrose 1963). We study algorithms for approximating the independence polynomial with negative and complex arguments. Whereas many algorithms for computing combinatorial polynomials are restricted to the univariate setting, we consider the multivariate independence polynomial, since there is a natural multivariate region of interest — Shearer’s region for the LLL. Our main result is: for any n-vertex graph of degree at most d, any α ∈ (0, 1], and any complex vector p such that (1 + α) · p lies in Shearer’s region, there is a deterministic algorithm to approximate the independence polynomial at p within (1 + ) multiplicative error and with runtime ( n α ) . Our results also extend to graphs of unbounded degree that have a bounded connective constant. Our analysis uses a novel multivariate form of the correlation decay technique. On the hardness side, we prove that every algorithm must have some dependence on α, as it is #P-hard to evaluate the polynomial within any poly (n) factor at an arbitrary given point in Shearer’s region. Similarly, deciding if a given point lies in Shearer’s region is also #P-hard. Finally, it is NP-hard to approximate the independence polynomial in a graph of degree at most d with a Vxed negative argument that is only a constant factor outside Shearer’s region. ∗Email: [email protected]. University of British Columbia. †Email: [email protected]. California Institute of Technology. Supported by NSF grant CCF-1319745. ‡Email: [email protected]. Stanford University. ar X iv :1 60 8. 02 28 2v 1 [ cs .D S] 7 A ug 2 01 6