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Deterministic approximate counting of colorings with fewer than $2Δ$ colors via absence of zeros

Ferenc Bencs, Khallil Berrekkal, Guus Regts

TL;DR

The paper proves a new zero-free region for the antiferromagnetic Potts model partition function $Z_G(q; w)$ that persists for all graphs with maximum degree $\Delta$ when $q\ge (2-\eta)\Delta$ with a fixed $\eta\ge 0.002$. This enables a deterministic polynomial-time algorithm, via Barvinok\'s interpolation, to approximate the number of proper $q$-colorings for such graphs, thereby breaking the longstanding $q=2\Delta$ barrier. The approach hinges on a careful analysis of log-ratios of colorings, induction on partially colored graphs, and gradient-based control of perturbations, augmented by refined local bounds on root marginals. The results also yield local central limit-type consequences for the number of monochromatic edges and open avenues for extensions to multivariate Potts, list-colorings, and small-degree regimes, with potential improvements for triangle-free graphs. Overall, the work advances deterministic approximate counting in graph coloring by leveraging a refined zero-free region and locality-aware inductive techniques.

Abstract

Let $Δ,q\geq 3$ be integers. We prove that there exists $η\geq 0.002$ such that if $q\geq (2-η)Δ$, then there exists an open set $\mathcal{U}\subset \mathbb{C}$ that contains the interval $[0,1]$ such that for each $w\in \mathcal{U}$ and any graph $G=(V,E)$ of maximum degree at most $Δ$, the partition function of the anti-ferromagnetic $q$-state Potts model evaluated at $w$ does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the $q=2Δ$-barrier for this problem. As a direct consequence we obtain via Barvinok's interpolation method a deterministic polynomial time algorithm to approximate the number of proper $q$-colorings of graphs of maximum degree at most $Δ$, provided $q\geq (2-η)Δ$.

Deterministic approximate counting of colorings with fewer than $2Δ$ colors via absence of zeros

TL;DR

The paper proves a new zero-free region for the antiferromagnetic Potts model partition function that persists for all graphs with maximum degree when with a fixed . This enables a deterministic polynomial-time algorithm, via Barvinok\'s interpolation, to approximate the number of proper -colorings for such graphs, thereby breaking the longstanding barrier. The approach hinges on a careful analysis of log-ratios of colorings, induction on partially colored graphs, and gradient-based control of perturbations, augmented by refined local bounds on root marginals. The results also yield local central limit-type consequences for the number of monochromatic edges and open avenues for extensions to multivariate Potts, list-colorings, and small-degree regimes, with potential improvements for triangle-free graphs. Overall, the work advances deterministic approximate counting in graph coloring by leveraging a refined zero-free region and locality-aware inductive techniques.

Abstract

Let be integers. We prove that there exists such that if , then there exists an open set that contains the interval such that for each and any graph of maximum degree at most , the partition function of the anti-ferromagnetic -state Potts model evaluated at does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the -barrier for this problem. As a direct consequence we obtain via Barvinok's interpolation method a deterministic polynomial time algorithm to approximate the number of proper -colorings of graphs of maximum degree at most , provided .
Paper Structure (21 sections, 17 theorems, 33 equations, 1 figure)

This paper contains 21 sections, 17 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The partition function of the Potts model.
  3. Absence of zeros in computer science.
  4. The 2Delta bound of Liu, Sinclair, and Srivastava LSS2Delta.
  5. Our contributions.
  6. Organization and conventions.
  7. Outline of approach and setup
  8. Definitions and expanded theorem statement
  9. Outline of proof and more definitions
  10. Basic bounds on marginal probabilities of the root vertex
  11. The behavior of F and nabla F
  12. Proof of main theorem for eta=0
  13. Refined bounds on marginal probabilities of the root vertex
  14. Local influences on the probabilities
  15. Proof of Theorem \ref{['thm:main expanded']} for eta>0
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists a constant $\eta\geq 0.002$ such that for all integers $\Delta\geq 3$ and $q\geq (2-\eta)\Delta$ there exists an open set $\mathcal{U}\subset \mathbb{C}$ containing the interval $[0,1]$ such that for each $w\in \mathcal{U}$ and graph $G$ of maximum degree at most $\Delta$, $Z_G(q,w)\neq

Figures (1)

  • Figure 4: Pictorial depiction of how the ratio $Z^{\ell_1}_{G,v}(w) /Z^{\ell_2}_{G,v}(w)$ is expressed as a telescoping product of the ratios $Z^{\ell_1}_{\hat{G}_i,\hat{v}_i}(w) / Z^{\ell_2}_{\hat{G}_i,\hat{v}_i}(w)$.

Theorems & Definitions (23)

  • Theorem 1.1
  • corollary 1.2
  • Theorem 2.1
  • remark 2.2
  • Lemma 2.3
  • Lemma 3.1
  • remark 3.2
  • Lemma 3.3
  • Lemma 4.1
  • proposition 4.2
  • ...and 13 more