Generating Hard Satisfiable Formulas by Hiding Solutions Deceptively

Abstract

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the ``hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to ``deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT.