Optimal myopic algorithms for random 3-SAT

Let F3(n,m) be a random 3-SAT formula formed by selecting uniformly, independently and with replacement, m clauses among all 8(nC3) possible 3-clauses over n variables. It has been conjectured that there exists a constant r3 such that, for any ε>0, F3[n,(r3-ε)n] is almost surely satisfiable, but F3[n,(r3+ε)n] is almost surely unsatisfiable. The best lower bounds for the potential value of r3 have come form analyzing rather simple extensions of unit-clause propagation. It was shown by D. Achlioptas (2000) that all these extensions can be cast in a common framework and analyzed in a uniform manner by employing differential equations. We determine optimal algorithms that are expressible in that framework, establishing r3 >3.26. We extend the analysis via differential equations, and make extensive use of a new optimization problem that we call the “max-density multiple-choice knapsack” problem. The structure of optimal knapsack solutions elegantly characterizes the choices made by an optimal algorithm.