The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem

TL;DR

This paper precisely determines the satisfiability threshold for a specific NP-complete random CSP, revealing its computational difficulty near the threshold and advancing understanding of phase transitions in complex problems.

Contribution

It is the first to identify an exact linear satisfiability threshold for a particular NP-complete random CSP model, similar to random k-SAT.

Findings

Exact satisfiability threshold determined

Instances near the threshold are computationally hard

Random instances with linear clauses have exponential resolution complexity

Abstract

We determine the exact threshold of satisfiability for random instances of a particular NP-complete constraint satisfaction problem (CSP). This is the first random CSP model for which we have determined a precise linear satisfiability threshold, and for which random instances with density near that threshold appear to be computationally difficult. More formally, it is the first random CSP model for which the satisfiability threshold is known and which shares the following characteristics with random k-SAT for k >= 3. The problem is NP-complete, the satisfiability threshold occurs when there is a linear number of clauses, and a uniformly random instance with a linear number of clauses asymptotically almost surely has exponential resolution complexity.