On the constraint length of random $k$-CSP

Abstract

Consider an instance $I$ of the random $k$-constraint satisfaction problem ($k$-CSP) with $n$ variables and $t=r\frac{n\ln d}{-\ln (1-p)}$ constraints, where $d$ is the domain size of each variable and $p$ determines the tightness of the constraints. Suppose that $d\ge 2$, $r>0$ and $0<p<1$ are constants, and $k\ge \alpha \ln n/\ln d$ for any fixed $\alpha >1/2$. We prove that

$$\begin{aligned} \nonumber \lim _{n\rightarrow \infty }\mathbf{ Pr } [I\ \text{ is } \text{ satisfiable }]=\left\{ \begin{array}{cc} 1 &{}\quad \text{ r } < 1, \\ 0 &{}\quad \text{ r } > 1. \\ \end{array} \right. \end{aligned}$$

Similar results also hold for the $k$-$hyper$-$\mathbf {F}$-$linear$ CSP which is obtained by incorporating certain algebraic structures to the domains and constraint relations of $k$-CSP.