Random 2-XORSAT at the Satisfiability Threshold

Abstract

We consider the random 2-XOR satisfiability problem, in which each instance is a formula that is a conjunction of m Boolean equations of the form x ⊕ y = 0 or x ⊕ y = 1. Random formulas on n Boolean variables are chosen uniformly at random from among all ${n(n-1)\choose m}$ possible choices. This problem is known to have a coarse transition as n and m tends to infinity in the ratio m/n →c in particular the probability p(n,cn) that a random 2-XOR formula is satisfiable tends to zero when c reaches c = 1/2. We determine the rate n ^− 1/12 at which this probability approaches zero inside the scaling window $m=\frac{n}{2}(1+\mu n^{-1/3})$. This main result is based on a first exact enumeration result about the number of connected components of some constrained family of random edge-weighted (0/1) graphs, namely those without cycles of odd weight. This study relies on the symbolic method and analytical tools coming from generating function theory which enable us to describe the evolution of $n^{1/12} \ p(n,\frac{n}{2}(1+\mu n^{-1/3}))$ as a function of μ. Our results are in accordance with those obtained by statistical physics methods, their tightness points out the benefit one could get in developping generating function methods for the investigation of phase transition associated to Constrained Satisfaction Problems.