Journal of Combinatorial Optimization

, Volume 30, Issue 1, pp 188–200 | Cite as

On the constraint length of random \(k\)-CSP

  • Guangyan Zhou
  • Zongsheng Gao
  • Jun Liu


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.


\(k\)-CSP The second moment method Threshold phenomena Phase transition 



The authors would like to thank the referees for their valuable suggestions. This research was supported by National Natural Science Fund of China (Grant No. 11171013, 11371225, 11301091).


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.LMIB and School of Mathematics and Systems ScienceBeihang UniversityBeijingChina

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