Critical Behavior in the Satisfiability of Random Boolean Expressions

Abstract

Determining the satisfiability of randomly generated Boolean expressions with k variables per clause is a popular test for the performance of search algorithms in artificial intelligence and computer science. It is known that for k = 2, formulas are almost always satisfiable when the ratio of clauses to variables is less than 1; for ratios larger than 1, the formulas are almost never satisfiable. Similar sharp threshold behavior is observed for higher values of k. Finite-size scaling, a method from statistical physics, can be used to characterize size-dependent effects near the threshold. A relationship can be drawn between thresholds and computational complexity.

Approximations and Randomization to Boost CSP Techniques

“…This is a new emerging area of research that is changing the way we characterize the computational complexity of NP-complete problems, beyond the worst-case complexity notion: Using tools from statistical physics we are now able to provide a fine characterization of the spectrum of computational complexity of instances of NP-complete problems, identifying typical easy-hard-easy patterns (Hogg and Huberman, 1996). For example, in the random Satisfiability problem, with fixed clause length (K-Sat, with K 3), it has been empirically shown that the difficulty of problems depends on the ratio between number of clauses and number of variables (Kirkpatrick and Selman, 1994). Furthermore, it has also been empirically shown that the peak in complexity occurs at the phase transition, i.e., the region in which instances change from being almost all solvable to being almost all unsolvable.…”

Sharp thresholds of graph properties, and the $k$-sat problem

“…The following problem has attracted much attention from physicists and computer scientists (see [20] for a survey on this topic): Let x 1 , x 2 , . .…”

Threshold values of random K‐SAT from the cavity method

“…This problem displays a very interesting threshold phenomenon when one takes the large N limit, keeping the ratio of clauses to variable, α = M/N , fixed. Numerical simulations [16] suggest the existence of a phase transition at a value α c (K) of this ratio: For α < α c (K) a randomly generated problem is satisfiable (SAT) with probability going to one in the large N limit, for α > α c (K) a randomly generated problem is not satisfiable (UNSAT) with probability going to one in the large N limit. This phase transition is particularly interesting because it turns out that the really difficult instances, from the algorithmic point of view, are those where α is close to α c .…”