Random MAX SAT, Random MAX CUT, and Their Phase Transitions

Abstract

Given a 2-SAT formula $F$ consisting of $n$ variables and $\cn$ random clauses, what is the largest number of clauses $\max F$ satisfiable by a single assignment of the variables? We bound the answer away from the trivial bounds of $(3/4)cn$ and $cn$. We prove that for $c<1$, the expected number of clauses satisfiable is $\cn-\Theta(1/n)$; for large $c$, it is $((3/4)c + \Theta(\sqrt{c}))n$; for $c = 1+\eps$, it is at least $(1+\eps-O(\eps^3))n$ and at most $(1+\eps-\Omega(\eps^3/\ln \eps))n$; and in the ``scaling window'' $c= 1+\Theta(n^{-1/3})$, it is $cn-\Theta(1)$. In particular, just as the decision problem undergoes a phase transition, our optimization problem also undergoes a phase transition at the same critical value $c=1$. Nearly all of our results are established without reference to the analogous propositions for decision 2-SAT, and as a byproduct we reproduce many of those results, including much of what is known about the 2-SAT scaling window. We consider ``online'' versions of MAX-2-SAT, and show that for one version, the obvious greedy algorithm is optimal. We can extend only our simplest MAX-2-SAT results to MAX-k-SAT, but we conjecture a ``MAX-k-SAT limiting function conjecture'' analogous to the folklore satisfiability threshold conjecture, but open even for $k=2$. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. Finally, for random MAXCUT (the size of a maximum cut in a sparse random graph) we prove analogous results.