Solution clustering in random satisfiability

Achlioptas, Dimitris

doi:10.1140/epjb/e2008-00324-5

Cited by 8 publications

(15 citation statements)

References 18 publications

(30 reference statements)

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“…However, here there are no corresponding rigorous hardness results, and since it is now known that polynomial time solvable problems like random XOR-SAT have the same type of clustering [6,7] this connection is no longer thought be straightforward. The solution clustering in itself has been verified for large k [8]. Another early product of applying the cavity method to random k-SAT is the survey-propagation algorithm.…”

Section: Introductionmentioning

confidence: 92%

Revisiting the cavity-method threshold for random 3-SAT

Lundow

Markström

2019

Phys. Rev. E

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A detailed Monte Carlo-study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies α3 ≤ 4.262. If the assumption is correct, this means that the actual threshold value for k = 3 is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large k. Our result thus indicate that there are distinct behaviours for k above and below some critical kc, and the cavity method may provide a correct mean-field picture for the range above kc.

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Section: Introductionmentioning

confidence: 92%

Revisiting the cavity-method threshold for random 3-SAT

Lundow

Markström

2019

Phys. Rev. E

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“…The ensemble picture and its similarity to spin-glass models, has attracted statistical physics methods from the theory of strongly disordered systems. In particular, replica symmetry and cavity methods made it possible to obtain a better description of the structure of the solution space [29][30][31][32][33][34][35][36][37][38]. Using these statistical ensemble methods, sharp transitions were found in the thermodynamic limit (N, M → ∞, α = const.)…”

Section: A Chaotic Transition In the Escape Ratementioning

confidence: 99%

“…as function of α. These include the clustering (or dynamical) transition point α d [34][35][36], the freezing transition α f [37,38], and the SAT/UNSAT satisfiability threshold α s [29] etc. It is in the range α ∈ [α f , α s ] where all known algorithms fail or take exponentially long to solve problems , however, recent numerical results indicate that backtracking survey propagation (BSP) can solve some problems efficiently within a range beyond the freezing transition, for 3-SAT [39].…”

Section: A Chaotic Transition In the Escape Ratementioning

confidence: 99%

Order-to-chaos transition in the hardness of random Boolean satisfiability problems

et al. 2016

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Transient chaos is a ubiquitous phenomenon characterizing the dynamics of phase-space trajectories evolving towards a steady-state attractor in physical systems as diverse as fluids, chemical reactions, and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems, and even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability (k-SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate κ, an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution. We show that for a given density of constraints and fixed number of Boolean variables N, the hardness of formulas in random k-SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints α, hardness appears through a second-order phase transition at α_{χ} in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such a transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter κ∼N^{B|α-α_{χ}|^{1-γ}} with 0<γ<1. We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints α is increased.

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“…Given the result from last section, using method from Achlioptas and Ricci-Tersenghi (referring to section 3 of [28], [10], section 3 of [29]), we actually have a concrete way to split the solution space and put solutions into different clusters:…”

Section: B Organization Of Clustersmentioning

confidence: 99%

“…According to [18], if BP equations is initialized in the planted solution then the fixed point is biased towards the planted solution above the reconstruction threshold r d , so FIG. 9 supports our calculation of r d .…”

Section: Experiments Using Belief Propagationmentioning

confidence: 99%

The solution space structure of random constraint satisfaction problems with growing domains

et al. 2015

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In this paper we study the solution space structure of model RB, a standard prototype of Constraint Satisfaction Problem (CSPs) with growing domains. Using rigorous the first and the second moment method, we show that in the solvable phase close to the satisfiability transition, solutions are clustered into exponential number of well-separated clusters, with each cluster contains sub-exponential number of solutions. As a consequence, the system has a clustering (dynamical) transition but no condensation transition. This picture of phase diagram is different from other classic random CSPs with fixed domain size, such as random K-Satisfiability (K-SAT) and graph coloring problems, where condensation transition exists and is distinct from satisfiability transition. Our result verifies some non-rigorous results obtained using cavity method from spin glass theory.PACS numbers: 89.75. Fb, 89.20.Ff

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Solution clustering in random satisfiability

Cited by 8 publications

References 18 publications

Revisiting the cavity-method threshold for random 3-SAT

Revisiting the cavity-method threshold for random 3-SAT

Order-to-chaos transition in the hardness of random Boolean satisfiability problems

The solution space structure of random constraint satisfaction problems with growing domains

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