Coloring Random Graphs

Mulet, Roberto; Pagnani, Andrea; Weigt, Martin; Zecchina, Riccardo

doi:10.1103/physrevlett.89.268701

Cited by 177 publications

(240 citation statements)

References 19 publications

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“…In Figure 11 we show for reference the case q = 3, which is somewhat more dubious because it is not clear yet to which class -from the replica theory point of view -it belongs. In any case, the value α * ∼ 2.275, far beyond α d = α K = 2, is quite close (but still smaller) to the one (α SP = 2.3) obtained in Ref [6] with the best algorithm -Survey Propagation (SP) [4]-for smaller sizes. Before concluding this section, we wish to compare for this case the present algorithm with the 'Belief Propagation' (BP) algorithm [9,30].…”

Section: Coloring Random Graphssupporting

confidence: 78%

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Landscape analysis of constraint satisfaction problems

2007

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We discuss an analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A 'benchmark' algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the 'clustering' and in some cases even the 'thermodynamic' transitions. This point has a simple geometric meaning and can be in principle determined with standard Statistical Mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy. We illustrate this for the graph three and four-coloring problem. For Packing problems the present discussion allows to better characterize the 'J-point', proposed as a systematic definition of Random Close Packing [1], and to place it in the context of other theories of glasses.

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Section: Coloring Random Graphssupporting

confidence: 78%

“…For the coloring problem, the col-uncol transition appears at α uncol (3) = 2.355 and α uncol (4) = 4.45 [6]. The other critical connectivities in the properties of the landscape were found and computed recently by [30].…”

Section: Dynamicmentioning

confidence: 93%

Landscape analysis of constraint satisfaction problems

2007

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“…For the statistical physics analysis of the q−coloring problem [30,32,44] we consider a Potts [25] spin model with anti-ferromagnetic interactions where each variable s i (spin, node, vertex) is in one of the q different states (colors) s = 1, . .…”

Section: A Definition Of the Modelmentioning

confidence: 99%

“…Recently, the existence of the clustered phase was proven rigorously in some cases for the satisfiability problem [36,37]. A major step was made by applying the cavity equations on a single instance: this led to the development of a very efficient messagepassing algorithm called Survey Propagation (SP) that was originally used for the satisfaction problem in [24] and later adapted for the coloring problem in [30]. Survey propagation allows one to find solutions of large random instances even in the clustered phase and very near to the coloring threshold.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in the coloring of random graphs

2007

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We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions).We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdős-Rényi and regular random graphs and determine their asymptotic values for large number of colors.Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

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“…We hope to obtain a better characterization of this resemblance as further research adds detail to the picture. If we succeed, it may be possible to obtain a generic prescription for determining all the points at which the chromatic number increases along the sequence, thus adding to what is already known [9,33].…”

Section: Graph Coloringmentioning

confidence: 99%

On the phase transitions of graph coloring and independent sets

Barbosa

Ferreira

2004

Physica A: Statistical Mechanics and its Applications

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We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are "shortest" peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the "widest" acyclic orientations and the independence number.

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Coloring Random Graphs

Cited by 177 publications

References 19 publications

Landscape analysis of constraint satisfaction problems

Landscape analysis of constraint satisfaction problems

Phase transitions in the coloring of random graphs

On the phase transitions of graph coloring and independent sets

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