Optimization of the dynamic transition in the continuous coloring problem

TL;DR

This paper develops theoretical tools to optimize the Hamiltonian in continuous coloring problems, analyzing phase transitions and comparing analytical predictions with simulations, to understand the problem's complexity and solution landscape.

Contribution

It introduces a systematic method to construct Hamiltonians that maximize the dynamical transition threshold in continuous constraint satisfaction problems.

Findings

The model exhibits a first-order phase transition for small angles.

Analytical predictions match Monte Carlo simulations for specific angle values.

The approach links continuous coloring to discrete coloring and MKK models.

Abstract

Random constraint satisfaction problems can exhibit a phase where the number of constraints per variable $\alpha$ makes the system solvable in theory on the one hand, but also makes the search for a solution hard, meaning that common algorithms such as Monte-Carlo method fail to find a solution. The onset of this hardness is deeply linked to the appearance of a dynamical phase transition where the phase space of the problem breaks into an exponential number of clusters. The exact position of this dynamical phase transition is not universal with respect to the details of the Hamiltonian one chooses to represent a given problem. In this paper, we develop some theoretical tools in order to find a systematic way to build a Hamiltonian that maximizes the dynamic $\alpha_{\rm d}$ threshold. To illustrate our techniques, we will concentrate on the problem of continuous coloring, where one tries to set an angle $x_i \in [0;2\pi]$ on each node of a network in such a way that no adjacent nodes are closer than some threshold angle $\theta$, that is $\cos(x_i - x_j) \leq \cos\theta$. This problem can be both seen as a continuous version of the discrete graph coloring problem or as a one-dimensional version of the the Mari-Krzakala-Kurchan (MKK) model. The relevance of this model stems from the fact that continuous constraint satisfaction problems on sparse random graphs remain largely unexplored in statistical physics. We show that for sufficiently small angle $\theta$ this model presents a random first order transition and compute the dynamical, condensation and Kesten-Stigum transitions; we also compare the analytical predictions with Monte Carlo simulations for values of $\theta = 2\pi/q$, $q \in \mathbb{N}$. Choosing such values of $q$ allows us to easily compare our results with the renowned problem of discrete coloring.