A sharp threshold for a random constraint satisfaction problem

We consider random instances I of a constraint satisfaction problem generalizing k-SAT: given n boolean variables, m ordered k-tuples of literals, and q “bad” clause assignments, find an assignment which does not set any of the k-tuples to a bad clause assignment. We consider the case where k = Ω(log n), and generate instance I by including every k-tuple of literals independently with probability p. Appropriate choice of the bad clause assignments results in random instances of k-SAT and notall-equal k-SAT. For constant q, a second moment method calculation yields the sharp threshold lim n→∞ Pr[I is satisfiable] = { 1, if p ≤ (1− ǫ) ln 2 qnk−1 ; 0, if p ≥ (1 + ǫ) ln 2 qnk−1 .