Publisher's Note: Communities of solutions in single solution clusters of a random<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi></mml:math>-satisfiability formula [Phys. Rev. E<b>80</b>, 066108 (2009)]

Zhou, Haijun; Ma, Hui

doi:10.1103/physreve.80.069903

Cited by 6 publications

(26 citation statements)

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“…According to our opinion, the main reason of this viscosity increase is the emergence of complex community structure in the solution cluster of the subformula F m as revealed by Ref. [15] (see Fig. 5).…”

Section: Simulation Results and Interpretationmentioning

confidence: 99%

“…[15,33], the connection pattern of a single solution cluster C m of a random K-SAT subformula F m may be quite heterogeneous. Some of the solutions may be densely inter-connected with each other but are only very sparsely connected to the other solutions of the same cluster.…”

Section: Simulation Results and Interpretationmentioning

confidence: 99%

“…[15,33] to study the solution space structure of single random K-SAT formulas. Here, based on the same idea of random walking, we present a simple stochastic solver SEQSAT for the random K-SAT problem.…”

Section: Random Walk Processesmentioning

confidence: 99%

“…However, the dynamics within such a solution cluster is not necessarily trivial. Simulations performed on random 3-SAT and 4-SAT formulas revealed that, as the constraint density α becomes relatively large, the structure of a single solution cluster of the formula is no longer homogeneous [15]. The solutions may aggregate into many communities, each of which containing a group of solutions that are densely connected to each other while the connections between two different communities are relatively sparse.…”

Section: Introductionmentioning

confidence: 99%

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

Zhou

2010

Eur. Phys. J. B

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Random K-satisfiability (K-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random K-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process SEQSAT, which satisfies a K-SAT formula in a sequential manner. Before satisfying each newly added clause, SEQSAT walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density α of the satisfied subformula is less than certain value αcm; however it slows down considerably as α > αcm and finally reaches a jammed state at α ≈ αj. The glassy dynamical behavior of SEQSAT for α ≥ αcm probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point αj is larger than the solution space clustering transition point α d , and its value can be predicted by a long-range frustration mean-field theory. For random K-SAT with K ≥ 4, however, our simulation results indicate that αj = α d . The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.

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Section: Simulation Results and Interpretationmentioning

confidence: 99%

Section: Simulation Results and Interpretationmentioning

confidence: 99%

Section: Random Walk Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhou

2010

Eur. Phys. J. B

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“…Ward's algorithm has been applied in many different fields ranging from RNA secondary structures over optimization problems to spin glasses [7,19,39,48]. It is an iterative procedure where initially each configuration comprises a single item cluster.…”

Section: A Hierarchical Clusteringmentioning

confidence: 99%

Numerical solution-space analysis of satisfiability problems

Mann

Hartmann

2010

Phys. Rev. E

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The solution-space structure of the 3-Satisfiability Problem (3-SAT) is studied as a function of the control parameter α (ratio of number of clauses to the number of variables) using numerical simulations. For this purpose, one has to sample the solution space with uniform weight. It is shown here that standard stochastic local-search (SLS) algorithms like "ASAT" and "MCMCMC" (also known as "parallel tempering") exhibit a sampling bias. Nevertheless, unbiased samples of solutions can be obtained using the "ballistic-networking approach", which is introduced here. It is a generalization of "ballistic search" methods and yields also a cluster structure of the solution space.As application, solutions of 3-SAT instances are generated using ASAT plus ballistic networking. The numerical results are compatible with a previous analytic prediction of a simple solution-space structure for small values of α and a transition to a clustered phase at α c ≈ 3.86, where the solution space breaks up into several non-negligible clusters. Furthermore, in the thermodynamic limit there are, for values of α close to the SAT-UNSAT transition α s ≈ 4.267, always clusters without any frozen variables. This may explain why some SLS algorithms are able to solve very large 3-SAT instances close to the SAT-UNSAT transition.

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Learning by random walks in the weight space of the Ising perceptron

Huang

Zhou

2010

J. Stat. Mech.

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Several variants of a stochastic local search process for constructing the synaptic weights of an Ising perceptron are studied. In this process, binary patterns are sequentially presented to the Ising perceptron and are then learned as the synaptic weight configuration is modified through a chain of single-or double-weight flips within the compatible weight configuration space of the earlier learned patterns. This process is able to reach a storage capacity of α ≈ 0.63 for pattern length N = 101 and α ≈ 0.41 for N = 1001. If in addition a relearning process is exploited, the learning performance is further improved to a storage capacity of α ≈ 0.80 for N = 101 and α ≈ 0.42 for N = 1001. We found that, for a given learning task, the solutions constructed by the random walk learning process are separated by a typical Hamming distance, which decreases with the constraint density α of the learning task; at a fixed value of α, the width of the Hamming distance distributions decreases with N .

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Publisher's Note: Communities of solutions in single solution clusters of a randomK-satisfiability formula [Phys. Rev. E80, 066108 (2009)]

Cited by 6 publications

References 0 publications

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

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

Numerical solution-space analysis of satisfiability problems

Learning by random walks in the weight space of the Ising perceptron

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