Hasso-Plattner-Institut25 Jahre HPI
Hasso-Plattner-Institut25 Jahre HPI

Ralf Rothenberger

Satisfiability Thresholds for Non-Uniform Random k-SAT

Boolean Satisfiability (SAT) is one of the problems at the core of theoretical computer science. It was the first problem proven to be NP-complete by Cook and, independently, by Levin. Nowadays it is conjectured that SAT cannot be solved in sub-exponential time. Thus, it is generally assumed that SAT and its restricted version k-SAT are hard to solve. However, state-of-the-art SAT solvers can solve even huge practical instances of these problems in a reasonable amount of time.

Why is SAT hard in theory, but easy in practice? One approach to answering this question is investigating the average runtime of SAT. In order to analyze this average runtime the random k-SAT model was introduced. The model generates all k-SAT instances with n variables and m clauses with uniform probability. Researching random k-SAT led to a multitude of insights and tools for analyzing random structures in general. One major observation was the emergence of the so-called satisfiability threshold: A phase transition point in the number of clauses at which the generated formulas go from asymptotically almost surely satisfiable to asymptotically almost surely unsatisfiable. Additionally, instances around the threshold seem to be particularly hard to solve.

In this thesis we analyze a more general model of random k-SAT that we call non-uniform random k-SAT. In contrast to the classical model each of the n Boolean variables now has a distinct probability of being drawn. For each of the m clauses we draw k variables according to the variable distribution and choose their signs uniformly at random. Non-uniform random k-SAT gives us more control over the distribution of Boolean variables in the resulting formulas. This allows us to tailor distributions to the ones observed in practice. Notably, non-uniform random k-SAT contains the previously proposed models random k-SAT, power-law random k-SAT and geometric random k-SAT as special cases.

We analyze the satisfiability threshold in non-uniform random k-SAT depending on the variable probability distribution. Our goal is to derive conditions on this distribution under which an equivalent of the satisfiability threshold conjecture holds. We start with the arguably simpler case of non-uniform random 2-SAT. For this model we show under which conditions a threshold exists, if it is sharp or coarse, and what the leading constant of the threshold function is. These are exactly the three ingredients one needs in order to prove or disprove the satisfiability threshold conjecture. For non-uniform random k-SAT with k≥3 we only prove sufficient conditions under which a threshold exists. We also show some properties of the variable probabilities under which the threshold is sharp in this case. These are the first results on the threshold behavior of non-uniform random k-SAT.