[Paper Review] Computing the Independence Polynomial: from the Tree Threshold down to the Roots

Nicholas J. A. Harvey, Piyush Srivastava+1 more|arXiv (Cornell University)|Aug 7, 2016

Markov Chains and Monte Carlo Methods44 references4 citations

TL;DR

This paper presents a deterministic algorithm to approximate the multivariate independence polynomial $ Z(\mathbf{z}) $ for complex and negative vertex activities in any root-free complex polydisc centered at the origin. Using a novel multivariate correlation decay technique with non-uniform complex parameters, it achieves a $(1+\epsilon)$-approximation in time $(n/\epsilon\alpha)^{O(\log d / \sqrt{\alpha})}$, unifying known computable regions and proving tightness of the $1/\sqrt{\alpha}$ dependence in the exponent.

ABSTRACT

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.

Motivation & Objective

• To develop a unified algorithm for approximating the independence polynomial $ Z(\mathbf{z}) $ across regions where it is computable, including negative and complex vertex activities.
• To extend Weitz’s correlation decay framework to handle non-uniform complex parameters, enabling approximation beyond the positive real regime.
• To establish tight runtime bounds for approximating $ Z(\mathbf{z}) $, particularly near critical thresholds where computational hardness arises.
• To provide efficient algorithms for testing membership in Shearer’s region and for algorithmic Lovász Local Lemma with slack $ \alpha $.
• To prove that approximating $ Z(\mathbf{z}) $ at arbitrary points in Shearer’s region and testing membership are #P-hard problems.

Proposed method

• The core method employs a novel multivariate form of correlation decay that handles non-uniform complex vertex activities, extending Weitz’s approach beyond positive reals.
• The algorithm operates within a root-free complex polydisc, ensuring $ Z(\mathbf{z}') \neq 0 $ for all $ |z'_i| \leq (1+\alpha)|z_i| $, guaranteeing analyticity and stability.
• A key component is the error sensitivity parameter, which bounds the decay of correlations in the computation tree, ensuring convergence to the correct value.
• The analysis uses complex analysis inequalities and recursive bounds on the recurrence $ f(x) = 1 - (1 - \lambda x)^d $, tracking convergence to fixed points.
• The runtime is derived from bounding the number of levels in the computation tree required for $ \epsilon $-accuracy, depending on $ \alpha $ and $ d $.
• Hardness results are established via reductions from known #P-hard problems, showing that approximation is hard even within Shearer’s region.

Experimental results

Research questions

• RQ1 Can Weitz’s correlation decay method be extended to handle complex and negative vertex activities in the independence polynomial?
• RQ2 What is the optimal dependence of the runtime on the slack parameter $ \alpha $ near the critical threshold for the independence polynomial?
• RQ3 Does the region where $ Z(\mathbf{z}) $ is approximately computable coincide with the region where it is non-zero (root-free)?
• RQ4 Can the algorithmic Lovász Local Lemma be implemented efficiently using polynomial evaluation and root-free regions?
• RQ5 Is testing membership in Shearer’s region or approximating $ Z(\mathbf{z}) $ within Shearer’s region computationally hard?

Key findings

• The algorithm achieves a $(1+\epsilon)$-approximation of the independence polynomial $ Z(\mathbf{z}) $ in time $ (n/\epsilon\alpha)^{O(\log d / \sqrt{\alpha})} $ for any complex $ \mathbf{z} $ in a root-free polydisc.
• The $ 1/\sqrt{\alpha} $ dependence in the exponent is optimal, as shown by a lower bound on the convergence rate of the recurrence $ f^l(0) $ near the critical point.
• For univariate real $ \lambda $, the algorithm works in the interval $ (-\lambda_c'(d), \lambda_c(d)) $, matching the known computational threshold where approximation becomes NP-hard outside.
• An algorithm to test membership in Shearer’s region within multiplicative error $ 1+\alpha $ runs in time $ (n/\alpha)^{O(\sqrt{n/\alpha} \log d)} $.
• A deterministic algorithm for Shearer’s lemma with $ n $ events and slack $ \alpha $ runs in time $ (nm/\alpha)^{O(\sqrt{m/\alpha} \log d)} $.
• Evaluating $ Z(\mathbf{z}) $ at arbitrary points in Shearer’s region and testing membership in Shearer’s region are both #P-hard problems.