Exhaustive enumeration unveils clustering and freezing in the random 3-satisfiability problem

Abstract

We study geometrical properties of the complete set of solutions of the random 3-satisfiability problem. We show that even for moderate system sizes the number of clusters corresponds surprisingly well with the theoretic asymptotic prediction. We locate the freezing transition in the space of solutions which has been conjectured to be relevant in explaining the onset of computational hardness in random constraint satisfaction problems. PACS numbers: 89.20.Ff,75.10.Nr,89.70.Eg Satisfiability (SAT) is one of the most important problems in theoretical computer science. It was the first problem shown to be NP-complete [1,2], and it is of central relevance in various practical applications, including artificial intelligence, planning, hardware and electronic design, automation, verification and more. It can thus be pictorially thought of as the Ising model of computer science. Ensembles of randomly generated SAT instances emerged in computer science as a way of evaluating algorithmic performance and addressing questions regarding the average case complexity.An instance of random K-SAT problem consists of N Boolean variables and M clauses. Each clause contains a subset of K distinct variables chosen uniformly at random, and each clause forbids one random assignment of the K variables out of the 2 K possible ones. The problem is satisfiable if there exists a variable assignment that simultaneously satisfies all clauses and we call such an assignments a solution to the problem. When the density of constraints α = M/N is increased, the formulas become less likely to be satisfiable. In the thermodynamical limit there is a sharp transition from a phase in which the formulas are almost surely satisfiable to a phase where they are almost surely unsatisfiable. The existence of this transition is partly established rigorously [3]. It is also a well known empirical result that the hardest instances are found near to this threshold [4,5,6].Random K-SAT has attracted interest of statistical physicists because of its equivalence to mean field spin glasses [7]. Indeed, the problem can be rephrased as minimizing a spin glass-like energy function which counts the number of violated clauses. The results and insights coming from this equivalence are remarkable. The satisfiability threshold and other phase transitions in the structure of solutions are described in [8,9,10]. In particular, it was shown that for K ≥ 3 the space of solutions for highly constrained but still satisfiable instances splits into exponentially many clusters and in some cases this clustering has been rigorously confirmed [11,12]. The so-called freezing of variables in clusters is another rich concept studied recently [13,14]. However, a detailed understanding of how the clustering or freezing of solutions affects the average computational hardness is still one of the most interesting open problems in the field.Since the exact statistical physics solution of the random satisfiability problem appeared [9,15] dozens of directly related articles followed. Mathemat...

A Better Algorithm for Random k-SAT

“…In fact, non-rigorous arguments as well as experimental evidence [5] suggest that the picture is quite different and rather more complicated for "small" k (say, k = 3, 4, 5). In this case the various phenomena that occur at (or very near) the point 2 k ln(k)/k for k ≥ 10 appear to happen at vastly different points in the satisfiable regime.…”

Glassy behavior and jamming of a random walk process for sequentially satisfying a constraint satisfaction formula

“…Later at the condensation transition point α = α c the solution space becomes dominated by only a few Gibbs states (for the special case of K = 3, the clustering and the condensation transition coincide). It was also found that some variables will become frozen to the same spin value in all the solutions of a Gibbs state [31,32], this fact has serious consequences for stochastic local search algorithms.…”

“…Our simulation results reveal that the jamming constraint densities α j as obtained for many trajectories of SEQSAT on the same random K-SAT formula and on different random K-SAT formulas are very close to each other. It is anticipated that, in the thermodynamic limit of N → ∞, the SEQSAT process has a true jamming transition at a critical constraint density α located at α = α f = 4.254 [32], while that for the random 4-SAT problem is located at α f = 9.88 [31]. These threshold constraint densities correspond to the appearance of frozen variables in the dominating Gibbs states of the solution space.…”

Numerical solution-space analysis of satisfiability problems

“…In a phase with a large number of small clusters we will miss small clusters if they only comprise a negligible part of the solutions (in the sense of the stopping criterion) and therefore underestimate the number of clusters. It is therefore natural that the complexity found here is lower than the one given in [26]. After all the complexity shown in the graph is only a lower bound for the true complexity respecting all clusters.…”

“…Therefore in [26] a different notion of frozen clusters via the whitening core is used. There one looks, for each solution, interatively for variables which can be flipped since they appear only in clauses satisfied by other variables or which contain variables already detected in the whitening core.…”