LATIN 2010: LATIN 2010: Theoretical Informatics pp 320-331

# Limit Theorems for Random MAX-2-XORSAT

• Vonjy Rasendrahasina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)

## 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.

## Keywords

MAX XORSAT Constraint Satisfaction Problem Phase transition Random graph Analytic Combinatorics

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