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

@inproceedings{Harvey2016ComputingTI, 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}, year={2016} }

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…

## 23 Citations

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- MathematicsArXiv
- 2022

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,…

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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

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- 2020

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

- Mathematics, Computer ScienceRandom Struct. Algorithms
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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

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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

- Mathematics, Computer ScienceJ. ACM
- 2019

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.

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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

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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.

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

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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.

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