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

  title={Computing the Independence Polynomial: from the Tree Threshold down to the Roots},
  author={Nicholas J. A. Harvey and Piyush Srivastava and Jan Vondr{\'a}k},
  booktitle={ACM-SIAM Symposium on Discrete Algorithms},
We study an algorithm for approximating the multivariate independence polynomial Z(z), with negative and complex arguments. While the focus so far has been mostly on computing combinatorial polynomials restricted to the univariate positive setting (with seminal results for the independence polynomial by Weitz (2006) and Sly (2010)), the independence polynomial with negative or complex arguments has strong connections to combinatorics and to statistical physics. The independence polynomial with… 

Figures from this paper

On the zeroes of hypergraph independence polynomials

We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms,

On complex roots of the independence polynomial

It is known from the work of Shearer (1985) (and also Scott and Sokal (2005)) that the independence polynomial $Z_G(\lambda)$ of a graph $G$ of maximum degree at most $d+1$ does not vanish provided

Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

It is shown that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold, improving the quasi-polynomial running time of Weitz's deterministic correlation decay algorithm.

Correlation decay and the absence of zeros property of partition functions

  • D. Gamarnik
  • Mathematics, Computer Science
    Random Struct. Algorithms
  • 2023
It is established that if the interpolation method developed by Barvinok is valid for a family of graphs, then this family exhibits a form of the correlation decay property which is asymptotic strong spatial mixing at superlogarithmic distances.

The Complexity of Approximating the Matching Polynomial in the Complex Plane

This work shows that for positive real $\gamma$ it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree) and explores whether the maximum degree parameter can be replaced by the connective Constant.

Approximate Counting, the Lovász Local Lemma, and Inference in Graphical Models

An algorithm to approximately count the number of solutions to a CNF formula Φ when the width is logarithmic in the maximum degree is introduced, which closes an exponential gap between the known upper and lower bounds.

Approximate counting, the Lovasz local lemma, and inference in graphical models

An algorithm to approximately count the number of solutions to a CNF formula Ф when the width is logarithmic in the maximum degree is introduced, which closes an exponential gap between the known upper and lower bounds.

Inapproximability of the independent set polynomial in the complex plane

The main result shows that for every λ outside of ΛΔ, the problem of approximating ZG(λ) on graphs G with maximum degree at most Δ is indeed NP-hard, and on the negative real axis, it is #P-hard to even decide whether ZG (λ)>0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak.

Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

This work introduces the contraction property as a unified sufficient condition to devise FPTAS via either Weitz's algorithm or Barvinok's algorithm and shows the existence of correlation decay in these regions based on the zero-freeness of the partition function.



Computing the independence polynomial in Shearer's region for the LLL

The multivariate independence polynomial is considered, since there is a natural multivariate region of interest -- Shearer's region for the LLL -- and its results extend to graphs of unbounded degree that have a bounded connective constant.

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the

Spatial Mixing and Approximation Algorithms for Graphs with Bounded Connective Constant

It is proved that thehard core model with vertex activity λ <; λc(Δ+1) exhibits strong spatial mixing on any graph of connective constant Δ, irrespective of its maximum degree, and hence an FPTAS is derived for the partition function of the hard core model on such graphs.

Approximating the Permanent of Graphs with Large Factors

Correlation decay and deterministic FPTAS for counting list-colorings of a graph

The principle insight of the present work is that the correlation decayproperty can be established with respect to certain computation tree, as opposed to the conventional correlation decay property which is typically established withrespect to graph theoretic neighborhoods of a given node.

Improved inapproximability results for counting independent sets in the hard‐core model

Sly's inapproximability result is extended by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs, which proves torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ.

Computing the partition function for graph homomorphisms with multiplicities

Spatial mixing and the connective constant: optimal bounds

The best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant are proved.

A constructive proof of the Lovász local lemma

This paper gives a randomized algorithm that finds a satisfying assignment to every k-CNF formula in which each clause has a neighbourhood of at most the asymptotic optimum of 2(k-5)-1 other clauses and that runs in expected time polynomial in the size of the formula, irrespective of k.

Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region

The first analog of the above inapproximability results for multi-spin systems is presented, and it is proved that for even k, in the tree non-uniqueness region, it is NP-hard, unless NP=RP, to approximate the number of colorings for triangle-free Δ-regular graphs.