On the Random Satisfiable Process

2006

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in N x variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2 , but they do not exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

Theoretical Computer Science, 2008

We study the structure of satisfying assignments of a random 3-Sat formula. In particular, we show that a random formula of density α ≥ 4.453 almost surely has no non-trivial "core" assignments. Core assignments are certain partial assignments that can be extended to satisfying assignments, and have been studied recently in connection with the Survey Propagation heuristic for random Sat. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-Sat with k ≥ 9 [1]. Our result implies that either this does not hold for 3-Sat or the threshold density for satisfiability in 3-Sat lies below 4.453. The main technical tool that we use is a novel simple application of the first moment method.

2005

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in N x variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2 , but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

SIAM Journal on Discrete Mathematics, 2011

It is known that random k-CNF formulas have a so-called satisfiability threshold at a density (namely, clause-variable ratio) of roughly 2 k ln 2: at densities slightly below this threshold almost all k-CNF formulas are satisfiable whereas slightly above this threshold almost no k-CNF formula is satisfiable. In the current work we consider satisfiable random formulas, and inspect another parameter-the diameter of the solution space (that is the maximal Hamming distance between a pair of satisfying assignments). It was previously shown that for all densities up to a density slightly below the satisfiability threshold the diameter is almost surely at least roughly n/2 (and n at much lower densities). At densities very much higher than the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment). In this paper we show that for all densities above a density that is slightly above the satisfiability threshold (more precisely at ratio (1 + ε)2 k ln 2, ε = ε(k) tending to 0 as k grows) the diameter is almost surely O(k2 −k n). This shows that a relatively small change in the density around the satisfiability threshold (a multiplicative (1 + ε) factor), makes a dramatic change in the diameter. This drop in the diameter cannot be attributed to the fact that a larger fraction of the formulas is not satisfiable (and hence have diameter 0), because the non-satisfiable formulas are excluded from consideration by our conditioning that the formula is satisfiable.

Algorithmica, 2016

This article is motivated by the following satisfiability question: pick uniformly at random an and{or Boolean expression of length n, built on a set of kn Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity?

Journal of Automated Reasoning, 2006

We consider a model for generating random k-SAT formulas, in which each literal occurs approximately the same number of times in the formula clauses (regular random k-SAT). Our experimental results show that such regular random k-SAT instances are much harder than the usual uniform random k-SAT problems. This is in agreement with other results that show that more balanced instances of random combinatorial problems are often much more difficult to solve than uniformly random instances, even at phase transition boundaries. There are almost no formal results known for such problem distributions. The balancing constraints add a dependency between variables that complicates a standard analysis. Regular random 3-SAT exhibits a phase transition as a function of the ratio α of clauses to variables. The transition takes place at approximately α = 3.5. We will show that for α > 3.78 w.h.p. 1 random Regular 3-SAT formulas are unsatisfiable. We will also show that the analysis of a greedy algorithm proposed in Kaporis et al (KKL02) for the uniform 3-SAT model can be adapted for regular random 3-SAT. In particular, we show that for formulas with ratio α < 2.46, a greedy algorithm finds a satisfying assignment with positive probability.

Theoretical Computer Science, 2008

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in N x variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2 , but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15, 2015

Random satisfiability problems with planted solutions exhibit an intriguing complexity gap: for problems on n variables with k variables per constraint, after O(n log n) random clauses the planted assignment becomes the unique solution but the best-known algorithms need at least max{n r/2 , n log n} to efficiently identify it (or even one that is correlated with it), for clause distributions that are (r − 1)-wise independent (thus r can be as high as k). We show a nearly tight unconditional lower bound ofΩ(max{n r/2 , n log n}) clauses for any statistical algorithma restricted class of algorithms introduced in [41, 29] that covers most algorithmic approaches commonly used in theory and practice. We complement this with a nearly matching upper bound: a simple, iterative, statistical algorithm that usesÕ(n r/2 ) clauses and time linear in this to find the planted assignment with high probability. As known approaches for planted satisfiability problems (spectral, MCMC, gradient-based, etc.) all have statistical analogues, this provides a rigorous explanation of the large gap between the identifiability and algorithmic identifiability thresholds for random satisfiability problems with planted solutions.

Boolean Models and Methods in Mathematics, Computer Science, and Engineering, 2009

Society for Industrial and Applied Mathematics eBooks, 2021

Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random k-SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k-SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. On the other hand, modeling locality with an underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time. A key ingredient for the result on geometric random k-SAT can be found in the complexity of higher-order Voronoi diagrams. As an additional technical contribution, we show an upper bound on the number of non-empty Voronoi regions, that holds for points with random positions in a very general setting. In particular, it covers arbitrary p-norms, higher dimensions, and weights affecting the area of influence of each point multiplicatively. Our bound is linear in the total weight. This is in stark contrast to quadratic lower bounds for the worst case.

Comput. Artif. Intell., 2001

In the last decade a lot of effort has been invested into both theoretical and experimental analysis of SAT phase transition. However, a deep theoretical understanding of this phenomenon is still lacking. Besides, many of experimental results are based on some assumptions that are not supported theoretically. In this paper we introduce the notion of SAT-equivalence and we prove that some restrictions often used in SAT experiments don&#39;t make an impact on location of a crossover point. We consider several fixed and random clause length SAT models and relations between them. We also discuss one new SAT model and report on a detected phase transition for it.

The solution space of a K-satisfiability (K-SAT) formula is a collection of solution clusters, each of which contains all the solutions that are mutually reachable through a sequence of singlespin flips. Knowledge of the statistical property of solution clusters is valuable for a complete understanding of the solution space structure and the computational complexity of the random K-SAT problem. This paper explores single solution clusters of random 3-and 4-SAT formulas through unbiased and biased random walk processes and the replica-symmetric cavity method of statistical physics. We find that the giant connected component of the solution space has already formed many different communities when the constraint density of the formula is still lower than the solution space clustering transition point. Solutions of the same community are more similar with each other and more densely connected with each other than with the other solutions. The entropy density of a solution community is calculated using belief propagation and is found to be different for different communities of the same cluster. When the constraint density is beyond the clustering transition point, the same behavior is observed for the solution clusters reached by several stochastic search algorithms. Taking together, the results of this work suggests a refined picture on the evolution of the solution space structure of the random K-SAT problem; they may also be helpful for designing new heuristic algorithms.

Encyclopedia of Algorithms, 2008

Random Structures & Algorithms, 2012

Let (C 1 , C 1), (C 2 , C 2),. .. , (C m , C m) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 n 2 clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on-line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on-line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999) 1/4. This contrasts with the well-known fact that a random m-clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as "Achlioptas processes." This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on-line setting above, we also consider an off-line version in which all m clause-pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off-line setting, we show that the two-choice satisfiability threshold for k-SAT for any fixed k coincides with the standard satisfiability threshold for random 2k-SAT.

, Schöning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n) · (4/3) n = O(1.334 n ). In this paper we present randomized algorithms and show that one of them has O(1.3302 n ) expected running time, improving Schöning's algorithm. (Note. At this point, the fastest randomized algorithm for 3-SAT is the one given by Iwama and Tamaki [IT03] that runs in O(1.324 n ).) *