How Frequently is a System of $2$-Linear Boolean Equations Solvable?

Abstract

We consider a random system of equations $x_i+x_j=b_{(i,j)} ({\rm mod }2)$, $(x_u\in \{0,1\},\, b_{(u,v)}=b_{(v,u)}\in\{0,1\})$, with the pairs $(i,j)$ from $E$, a symmetric subset of $[n]\times [n]$. $E$ is chosen uniformly at random among all such subsets of a given cardinality $m$; alternatively $(i,j)\in E$ with a given probability $p$, independently of all other pairs. Also, given $E$, ${\rm Pr}\{b_{e}=0\}={\rm Pr}\{b_e=1\}$ for each $e\in E$, independently of all other $b_{e\prime}$. It is well known that, as $m$ passes through $n/2$ ($p$ passes through $1/n$, resp.), the underlying random graph $G(n,\#{\rm edges}=m)$, ($G(n,{\rm Pr}({\rm edge})=p)$, resp.) undergoes a rapid transition, from essentially a forest of many small trees to a graph with one large, multicyclic, component in a sea of small tree components. We should expect then that the solvability probability decreases precipitously in the vicinity of $m\sim n/2$ ($p\sim 1/n$), and indeed this probability is of order $(1-2m/n)^{1/4}$, for $m < n/2$ ($(1-pn)^{1/4}$, for $p < 1/n$, resp.). We show that in a near-critical phase $m=(n/2)(1+\lambda n^{-1/3})$ ($p=(1+\lambda n^{-1/3})/n$, resp.), $\lambda=o(n^{1/12})$, the system is solvable with probability asymptotic to $c(\lambda)n^{-1/12}$, for some explicit function $c(\lambda)>0$. Mike Molloy noticed that the Boolean system with $b_e\equiv 1$ is solvable iff the underlying graph is $2$-colorable, and asked whether this connection might be used to determine an order of probability of $2$-colorability in the near-critical case. We answer Molloy's question affirmatively and show that, for $\lambda=o(n^{1/12})$, the probability of $2$-colorability is ${}\lesssim 2^{-1/4}e^{1/8}c(\lambda)n^{-1/12}$, and asymptotic to $2^{-1/4}e^{1/8}c(\lambda)n^{-1/12}$ at a critical phase $\lambda=O(1)$, and for $\lambda\to -\infty$.