Number of Guards Needed by a Museum: A Phase Transition in Vertex Covering of Random Graphs

Weigt, Martin; Hartmann, Alexander K.

doi:10.1103/physrevlett.84.6118

Cited by 173 publications

(264 citation statements)

References 13 publications

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“…The results obtained by SP and those obtained by the mean-field statistical physics calculations are in very good agreement. At given vertex degree c, the minimal VC density estimated by SP and the mean-field theory is lower than the corresponding value obtained through exact enumeration followed by extrapolation [17]. The reason for such a discrepancy can be understood.…”

Section: B Vc Density and Complexitymentioning

confidence: 81%

“…IV B we have seen, that it is possible to average the solution of warning propagation over random graphs of average vertex degree c, and to recover the replica-symmetric results of [17] in the thermodynamic limit. In analogy, the equations of survey propagation can be used to reproduce and extend the results of [19] on the typical properties of minimal VCs under the assumption of one step of replica-symmetry breaking (1RSB), i.e., to translate the probabilistic-algorithmic approach on single graph instances to a statistical-physics approach with the graph randomness playing the role of the quenched disorder.…”

Section: From Survey Propagation To Typical Properties On Random mentioning

confidence: 97%

“…The typical properties of VCs on such graphs have already been analyzed both with rigorous mathematical tools [14,15] and with statistical-physics methods [16,17,18,19,20]. We therefore have a pretty complete knowledge about the phase diagram of this problem, and can systematically compare the algorithmic performance of the message-passing techniques on single, finite, randomly generated graphs to the average behavior in the thermodynamic limit.…”

Section: Introductionmentioning

confidence: 99%

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Message passing for vertex covers

2006

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Constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints. In this paper, we develop and analyze message passing techniques, namely warning and survey propagation, which serve as efficient heuristic algorithms for solving these computational hard problems. We show also, how previously obtained results on the typical-case behavior of vertex covers of random graphs can be recovered starting from the message passing equations, and how they can be extended.

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Section: B Vc Density and Complexitymentioning

confidence: 81%

Section: From Survey Propagation To Typical Properties On Random mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Message passing for vertex covers

2006

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“…The most striking results so far were obtained in the description of the solution-space structure of the random satisfiability problem [3,4,5,6,7], of the number partitioning problem [8,9], of vertex covers [10,11,12] or colorings [13] of random graphs. In these cases, equilibrium methods from statistical mechanics can be applied directly, including e.g.…”

Section: Introductionmentioning

confidence: 99%

Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

2003

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Stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems. We give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems. We find two different dynamical regimes, depending on the number of constraints per variable: For low constraintness, the problems are solved efficiently, i.e. in linear time. For higher constraintness, the solution times become exponential. We observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium. If the algorithm runs long enough, an exponentially rare fluctuation towards a solution appears.

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“…In particular, one is interested in the question, how the change of the solution-space structures has an influence on the performance of exact and stochastic algorithms. For example, for the vertex-cover problem, which is defined on graphs, a clustering transition has been found analytically [6] and numerically [7,8] when increasing the edge density of Erdős-Rényi random graphs. This transition coincides with a change of the typical-case complexity from polynomial to exponential [9].…”

Section: Introductionmentioning

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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Number of Guards Needed by a Museum: A Phase Transition in Vertex Covering of Random Graphs

Cited by 173 publications

References 13 publications

Message passing for vertex covers

Message passing for vertex covers

Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

Numerical solution-space analysis of satisfiability problems

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