Computational Transition at the Uniqueness Threshold

TL;DR

This paper proves that approximating the partition function of the hardcore model on graphs of maximum degree d becomes computationally hard at the uniqueness threshold, confirming a key conjecture linking phase transitions and computational complexity.

Contribution

It establishes the exact computational threshold at the phase transition for the hardcore model, linking statistical physics and computational hardness rigorously.

Findings

Approximate counting becomes NP-hard at the uniqueness threshold.

The computational threshold coincides with the statistical physics phase transition.

No polynomial-time algorithm exists for counting independent sets at degree 6.

Abstract

The hardcore model is a model of lattice gas systems which has received much attention in statistical physics, probability theory and theoretical computer science. It is the probability distribution over independent sets $I$ of a graph weighted proportionally to $\lambda^{|I|}$ with fugacity parameter $\lambda$. We prove that at the uniqueness threshold of the hardcore model on the $d$-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree $d$. Specifically, we show that unless NP$=$RP there is no polynomial time approximation scheme for the partition function (the sum of such weighted independent sets) on graphs of maximum degree $d$ for fugacity $\lambda_c(d) < \lambda < \lambda_c(d) + \epsilon(d)$ where $\lambda_c = \frac{(d-1)^{d-1}}{(d-2)^d}$ is the uniqueness threshold on the $d$-regular tree and $\epsilon(d)>0$. Weitz produced an FPTAS for approximating the partition function when $0<\lambda < \lambda_c(d)$ so this result demonstrates that the computational threshold exactly coincides with the statistical physics phase transition thus confirming the main conjecture of [28]. We further analyze the special case of $\lambda=1, d=6$ and show there is no polynomial time algorithm for approximately counting independent sets on graphs of maximum degree $d= 6$ which is optimal. Our proof is based on specially constructed random bi-partite graphs which act as gadgets in a reduction to MAX-CUT. Building on the second moment method analysis of [28] and combined with an analysis of the reconstruction problem on the tree our proof establishes a strong version of 'replica' method heuristics developed by theoretical physicists. The result establishes the first rigorous correspondence between the hardness of approximate counting and sampling with statistical physics phase transitions.