7 research outputs found
Solution space heterogeneity of the random K-satisfiability problem: Theory and simulations
The random K-satisfiability (K-SAT) problem is an important problem for
studying typical-case complexity of NP-complete combinatorial satisfaction; it
is also a representative model of finite-connectivity spin-glasses. In this
paper we review our recent efforts on the solution space fine structures of the
random K-SAT problem. A heterogeneity transition is predicted to occur in the
solution space as the constraint density alpha reaches a critical value
alpha_cm. This transition marks the emergency of exponentially many solution
communities in the solution space. After the heterogeneity transition the
solution space is still ergodic until alpha reaches a larger threshold value
alpha_d, at which the solution communities disconnect from each other to become
different solution clusters (ergodicity-breaking). The existence of solution
communities in the solution space is confirmed by numerical simulations of
solution space random walking, and the effect of solution space heterogeneity
on a stochastic local search algorithm SEQSAT, which performs a random walk of
single-spin flips, is investigated. The relevance of this work to glassy
dynamics studies is briefly mentioned.Comment: 11 pages, 4 figures. Final version as will appear in Journal of
Physics: Conference Series (Proceedings of the International Workshop on
Statistical-Mechanical Informatics, March 7-10, 2010, Kyoto, Japan
Exhaustive enumeration unveils clustering and freezing in random 3-SAT
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.Comment: 4 pages, 3 figure
Solution to Satisfiability problem by a complete Grover search with trapped ions
The main idea in the original Grover search (Phys. Rev. Lett. 79, 325 (1997))
is to single out a target state containing the solution to a search problem by
amplifying the amplitude of the state, following the Oracle's job, i.e., a
black box giving us information about the target state. We design quantum
circuits to accomplish a complete Grover search involving both the Oracle's job
and the amplification of the target state, which are employed to solve
Satisfiability (SAT) problems. We explore how to carry out the quantum circuits
by currently available ion-trap quantum computing technology.Comment: 14 pages, 6 figure
Clusters of solutions and replica symmetry breaking in random k-satisfiability
We study the set of solutions of random k-satisfiability formulae through the
cavity method. It is known that, for an interval of the clause-to-variables
ratio, this decomposes into an exponential number of pure states (clusters). We
refine substantially this picture by: (i) determining the precise location of
the clustering transition; (ii) uncovering a second `condensation' phase
transition in the structure of the solution set for k larger or equal than 4.
These results both follow from computing the large deviation rate of the
internal entropy of pure states. From a technical point of view our main
contributions are a simplified version of the cavity formalism for special
values of the Parisi replica symmetry breaking parameter m (in particular for
m=1 via a correspondence with the tree reconstruction problem) and new large-k
expansions.Comment: 30 pages, 14 figures, typos corrected, discussion of appendix C
expanded with a new figur
On the thresholds in linear and nonlinear Boolean equations
We introduce a generalized XORSAT model, named as Massive Algebraic System (Hereafter abbreviated as MAS) consisting of linear and nonlinear Boolean equations. Through adjusting the proportion of nonlinear equations, denoted by p, this MAS model smoothly interpolates between XORSAT (p = 0) and MAS-nonlinear (p = 1). We conduct a systematic and complete study about a series of phase transitions in the space of solutions at given p and also present how the phase diagram evolves with the increase of p. First of all, using the probabilistic method and energetic 1RSB cavity method, we compute the satisfiability thresholds for any given p â [0,1) and determine a region where the satisfaction of problem all depends on its subproblem MAS-nonlinear. Furthermore, we locate three important non-satisfiability transitions, i.e. clustering, condensation and freezing, using entropic 1RSB cavity method, and find the space of solution undergoing different phase transition processes with the increase of p