A polynomial time computation of the exact correlation structure of k-satisfiability landscapes

The autocorrelation function and related correlation length are statistical quantities that capture the ruggedness of the fitness landscape: a measure that is directly related to the hardness of a problem for certain heuristic search algorithms. Typically, these quantities are estimated empirically by sampling along a random walk. In this paper, we show that a polynomial-time Walsh decomposition of the k-satisfiability evaluation function allows us to compute the exact autocorrelation function and correlation length for any given k-satisfiability instance. We also use the decomposition to compute a theoretical expectation for the autocorrelation function and correlation length over the ensemble of instances generated uniformly at random. We find that this expectation is invariant to the constrainedness of the problem as measured by the ratio of clauses to variables. However, we show that filtered problems, which are typically used in local search studies, have a bias that causes a significant deviation from the expected correlation structure of unfiltered, uniformly generated problems.