Zeros, chaotic ratios and the computational complexity of approximating the independence polynomial

de Boer, David; Buys, Pjotr; Guerini, Lorenzo; Peters, Han; Regts, Guus

doi:10.48550/arxiv.2104.11615

Cited by 4 publications

(8 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An important tool in our proof of Theorem 2 is the use of Montel's theorem. This is a cornerstone result in the theory of modern complex dynamical systems [12,32,41] and has recently found applications in the study of determining the location of zeros and the complexity of approximating evaluations of the independence polynomial [9,11,15].…”

Section: Montel's Theoremmentioning

confidence: 95%

Absence of zeros implies strong spatial mixing

Regts

2023

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

In this paper we show that absence of complex zeros of the partition function of the hard-core model on any family of bounded degree graphs that is closed under taking induced subgraphs implies that the associated probability measure, the hard-core measure, satisfies strong spatial mixing on that family. As a corollary we obtain that the hard-core measure on the family of bounded degree claw-free graphs satisfies strong spatial mixing for every value of the fugacity parameter. We furthermore derive strong spatial mixing for graph homomorphism measures from absence of zeros of the graph homomorphism partition function.

show abstract

Section: Montel's Theoremmentioning

confidence: 95%

Absence of zeros implies strong spatial mixing

Regts

2023

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

show abstract

“…Here we describe a few more recent works in this direction. Recent work of de Boer, Buys, Guerini, Peters, and Regts [13] establishes strong formal connections between the computational complexity of the hard core model, complex dynamics, and zero-freeness of the partition function (see Main Theorem of [13]): in particular they prove that the zeros of Z G for graphs in G d+1 are dense in the complement of U d . The limit shapes of the zero-free regions have also been studied: Bencs, Buys, and Peters [8] show that in the d → ∞ limit, a rescaled version of zero-free region tends to a bounded 0-star shaped region, whose boundary intersects lim d→∞ d • ∂U d only at real parameters.…”

Section: Contributionsmentioning

confidence: 96%

On complex roots of the independence polynomial

Bencs¹,

Csíkvári²,

Srivastava³

et al. 2023

Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

View full text Add to dashboard Cite

The independence polynomial of a graph is the generating polynomial of all its independent sets. Formally, given a graph G, its independence polynomial ZG(λ) is given by I λ |I| , where the sum is over all independent sets I of G. The independence polynomial has been an important object of study in both combinatorics and computer science. In particular, the algorithmic problem of estimating ZG(λ) for a fixed positive λ on an input graph G is a natural generalization of the problem of counting independent sets, and its study has led to some of the most striking connections between computational complexity and the theory of phase transitions. More surprisingly, the independence polynomial for negative and complex values of λ also turns out to be related to problems in statistical physics and combinatorics. In particular, the locations of the complex roots of the independence polynomial of bounded degree graphs turn out to be very closely related to the Lovász local lemma, and also to the questions in the computational complexity of counting. Consequently, the locations of such zeros have been studied in many works. In this direction, it is known from the work of Shearer [29] and of Scott and Sokal [27] -inspired by the study of the Lovász local lemma -that the independence polynomial ZG(λ) of a graph G of maximum degree at most d + 1 does not vanish provided that |λ| ≤ d d (d+1) d+1 . Significant extensions of this result have recently been given in the case when λ is in the right half-plane (i.e., when ℜλ ≥ 0) by Peters and Regts [26] and Bencs and Csikvári [9]. In this paper, our motivation is to further extend these results to find new zero free regions not only in the right half plane, but also in the left half-plane, that is, when ℜλ ≤ 0.We give new geometric criterions for establishing zero-free regions as well as for carrying out semi-rigorous numerical explorations. We then provide two examples of the (rigorous) use of these criterions, by establishing two new zero-free regions in the left-half plane. We also extend the results of Bencs and Csikvári [9] for the right half-plane using our framework. By a direct application of the interpolation method of Barvinok [5], combined with extensions due to Patel and Regts [25], our results also imply deterministic polynomial time approximation algorithms for the independence polynomial of bounded degree graphs in the new zero-free regions.

show abstract

“…Finally we make a conjecture about the zero-free locus of independence polynomials of boundeddegree hypergraphs. Using the notation from, e.g., [16,21,6], let U ∆ denote the maximal simply connected open set in C containing 0 that is zero-free for independence polynomials of all graphs of maximum-degree ∆. Extending this notation, let U ∆,k be the same for k-uniform hypergraphs (so U ∆ = U ∆,2 ); and U ∆,≥k the same for hypergraphs with edge-size at least k. In particular, Theorem 1 shows that U ∆,≥2 contains a disk of radius λ s (∆ + 1).…”

Section: Zero-freeness For Hypergraph Independence Polynomialsmentioning

confidence: 99%

“…Given the zero-freeness result of Theorem 1, we can obtain an FPTAS for Z G (λ) and prove Theorem 7 following Barvinok's method of polynomial interpolation: truncating the Taylor series for log Z G (λ) (in fact, the cluster expansion) around 0 after a given number of terms. This approach has been used in several recent works on approximate counting, including [46,48,32,7,12,21] on approximating the independence polynomial of bounded-degree graphs for (possibly complex) values of λ.…”

Section: Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

On the zeroes of hypergraph independence polynomials

Galvin¹,

McKinley²,

Perkins³

et al. 2022

Preprint

View full text Add to dashboard Cite

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, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree ∆ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree ∆ established by Shearer (of radius ∼ 1/(e∆)). Up to logarithmic factors in ∆ this is optimal, even for hypergraphs with all edge-sizes strictly greater than 2. We conjecture that for k ≥ 3, k-uniform linear hypergraphs have a much larger zero-free disk of radius Ω(∆ − 1 k−1 ). We establish this in the case of linear hypertrees.

show abstract

Zeros, chaotic ratios and the computational complexity of approximating the independence polynomial

Cited by 4 publications

References 20 publications

Absence of zeros implies strong spatial mixing

Absence of zeros implies strong spatial mixing

On complex roots of the independence polynomial

On the zeroes of hypergraph independence polynomials

Contact Info

Product

Resources

About