Limit Theorems for Random MAX-2-XORSAT

Abstract

We consider random instances of the MAX-2-XORSAT optimization problem. A 2-XOR formula is a conjunction of Boolean equations (or clauses) of the form x ⊕ y = 0 or x ⊕ y = 1. The MAX-2-XORSAT problem asks for the maximum number of clauses which can be satisfied by any assignment of the variables in a 2-XOR formula. In this work, formula of size m on n Boolean variables are chosen uniformly at random from among all $\binom{n(n-1) }{ m}$ possible choices. Denote by X _n,m the minimum number of clauses that can not be satisfied in a formula with n variables and m clauses. We give precise characterizations of the r.v. X _n,m around the critical density $\frac{m}{n} \sim \frac{1}{2}$ of random 2-XOR formula. We prove that for random formulas with m clauses X _n,m converges to a Poisson r.v. with mean $-\frac{1}{4}\log(1-2c)-\frac{c}{2}$ when m = cn, c ∈ ]0,1/2[ constant. If $m= \frac{n}{2}-\frac{\mu}{2}n^{2/3}$, μ and n are both large but μ = o(n ^1/3), $\frac{X_{n,m}-\lambda} {\sqrt{\lambda}}$ with $\lambda=\frac{\log{n}}{12} -\frac{\log{\mu}}{4}$ is normal. If $m = \frac{n}{2} + O(1)n^{2/3}$, $\frac{X_{n,m}- \frac{\log{n}}{12}}{\sqrt{\frac{\log{n}}{12}}}$ is normal. If $m = \frac{n}{2} + \frac{\mu}{2}n^{2/3}$ with 1 ≪ μ = o(n ^1/3) then $\frac{ 12X_{n,m}}{2\mu^3+\log{n}-3\log(\mu)} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1$. For any absolute constant ε> 0, if μ = εn ^1/3 then $\frac{8(1+\varepsilon)}{n( \varepsilon^2 - \sigma^2)} X_{n,m} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1$ where σ ∈ (0,1) is the solution of (1 + ε)e ^− ε = (1 − σ)e ^σ. Thus, our findings describe phase transitions in the optimization context similar to those encountered in decision problems.