Relaxation in graph coloring and satisfiability problems

Svenson, Pontus; Nordahl, Mats G.

doi:10.1103/physreve.59.3983

Cited by 44 publications

(16 citation statements)

References 54 publications

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“…thus allowing a non-zero solution for ∆ 1 to develop. This value is in surprisingly good agreement with a critical slowing down found numerically by Svenson and Nordahl [17]. They considered a simple zero-temperature Glauber dynamics for random satisfiability and coloring problems.…”

Section: The Continuous Transitionsupporting

confidence: 87%

A variational description of the ground state structure in random satisfiability problems

2000

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A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica symmetric results and also allows for the inclusion of replica symmetry breaking effects. For the 3-SAT problem, we find two transitions as the ratio α of logical clauses per Boolean variables increases. At the first one αs ≃ 3.96, a non-trivial organization of the solution space in geometrically separated clusters emerges. The multiplicity of these clusters as well as the typical distances between different solutions are calculated. At the second threshold αc ≃ 4.48, satisfying assignments disappear and a finite fraction B0 ≃ 0.13 of variables are overconstrained and take the same values in all optimal (though unsatisfying) assignments. These values have to be compared to αc ≃ 4.27, B0 ≃ 0.4 obtained from numerical experiments on small instances. Within the present variational approach, the SAT-UNSAT transition naturally appears as a mixture of a first and a second order transition. For the mixed 2+p-SAT with p < 2/5, the behavior is as expected much simpler: a unique smooth transition from SAT to UNSAT takes place at αc = 1/(1 − p).

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Section: The Continuous Transitionsupporting

confidence: 87%

A variational description of the ground state structure in random satisfiability problems

2000

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“…This is the 'threshold' energy density e th where a simple zero temperature Metropolis algorithm (ZTMA) will be trapped. This implies that ZTMA should find satisfying assignments only for α < α D , in agreement with the numerical results of [39]. These predictions can be tested most clearly through their generalization to single instances which we discuss in the next section.…”

Section: E Phase Diagram Of the Random 3sat Problemsupporting

confidence: 76%

“…This expression depends on the precise spin which has been added through the choice of the distributions P ℓ and P ′ ℓ , and of couplings J ℓ , which appear in (39). As one expects Σ(α, ǫ) to be self-averaging, one must average the logarithm of the expressions in (41) over the iteration of population dynamics algorithm.…”

Section: Computing the Energy And The Complexitymentioning

confidence: 99%

RandomK-satisfiability problem: From an analytic solution to an efficient algorithm

2002

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We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the 3-sat problem, with very good performances.

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“…Its precise relationship with dynamical properties, and in particular with the computational cost for finding a solution is not fully elucidated yet [21,22,28].…”

Section: Definitions Known Results and Phenomenologymentioning

confidence: 99%

Relaxation and metastability in a local search procedure for the random satisfiability problem

Semerjian

Monasson

2003

Phys. Rev. E

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An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio α of constraints per variable, resolution takes a time Tres growing linearly (Tres ∼ τres(α) N, α < α d ) or exponentially (Tres ∼ exp(N ζ(α)), α > α d ) with the size N of the instance. The relaxation time τres(α) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For α > α d , the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height ζ(α) of this barrier is proposed. The polynomial/exponentiel cross-over α d is not related to the onset of clustering among solutions.

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Relaxation in graph coloring and satisfiability problems

Cited by 44 publications

References 54 publications

A variational description of the ground state structure in random satisfiability problems

A variational description of the ground state structure in random satisfiability problems

RandomK-satisfiability problem: From an analytic solution to an efficient algorithm

Relaxation and metastability in a local search procedure for the random satisfiability problem

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