The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic

We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The analysis is carried out by mapping the history of the search process onto an out-of-equilibrium (multi-dimensional) surface growth problem. The growth exponent of the average running time is quantitatively predicted, in agreement with simulations.Comment: 5 figure

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA

Bicoloring Random Hypergraphs

We study the problem of bicoloring random hypergraphs, both numerically and analytically. We apply the zero-temperature cavity method to find analytical results for the phase transitions (dynamic and static) in the 1RSB approximation. These points appear to be in agreement with the results of the numerical algorithm. In the second part, we implement and test the Survey Propagation algorithm for specific bicoloring instances in the so called HARD-SAT phase.Comment: 14 pages, 10 figure

Optimization hardness as transient chaos in an analog approach to constraint satisfaction

Boolean satisfiability [1] (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for $k\geq 3$) implies efficient solutions to a large number of hard optimization problems [2,3]. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic [4-7], and the boundaries between the basins of attraction [8] of the solution clusters become fractal [7-9], signaling the appearance of optimization hardness [10]. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT [11] and of locked occupation problems [12] (considered among the hardest algorithmic benchmarks); a property partly due to the system's hyperbolic [4,13] character. The system finds solutions in polynomial continuous-time, however, at the expense of exponential fluctuations in its energy function.Comment: 27 pages, 14 figure

Sampling grid colourings with fewer colours

http://www.springerlink.com/content/hxn3ubgp4q7yp51j/ The original publication can be found at www.springerlink.comWe provide an optimally mixing Markov chain for 6-colorings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colorings has strong spatial mixing. Four and five are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colorings of the square grid