Limit theorems for random MAX-2-XORSAT

• Authors: Vonjy Rasendrahasina;Vlady Ravelomanana
• Venue: LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
• Year: 2010

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 Xn,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. Xn,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 Xn,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(n1/3), $\frac{X_{n,m}-\lambda} {\sqrt{\lambda}}$ 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(n1/3) then $\frac{ 12X_{n,m}}{2\mu^3+\log{n}-3\log(\mu)} {\mathbin{\stackrel{{\mathop{\mathrm{dist.}}}}{\longrightarrow}}} 1$. For any absolute constant ε0, if μ=εn1/3 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)eσ. Thus, our findings describe phase transitions in the optimization context similar to those encountered in decision problems.