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

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

A Curve Shaped Description of Large Networks, with an Application to the Evaluation of Network Models

BACKGROUND: Understanding the structure of complex networks is a continuing challenge, which calls for novel approaches and models to capture their structure and reveal the mechanisms that shape the networks. Although various topological measures, such as degree distributions or clustering coefficients, have been proposed to characterize network structure from many different angles, a comprehensive and intuitive representation of large networks that allows quantitative analysis is still difficult to achieve. METHODOLOGY/PRINCIPAL FINDINGS: Here we propose a mesoscopic description of large networks which associates networks of different structures with a set of particular curves, using breadth-first search. After deriving the expressions of the curves of the random graphs and a small-world-like network, we found that the curves possess a number of network properties together, including the size of the giant component and the local clustering. Besides, the curve can also be used to evaluate the fit of network models to real-world networks. We describe a simple evaluation method based on the curve and apply it to the Drosophila melanogaster protein interaction network. The evaluation method effectively identifies which model better reproduces the topology of the real network among the given models and help infer the underlying growth mechanisms of the Drosophila network. CONCLUSIONS/SIGNIFICANCE: This curve-shaped description of large networks offers a wealth of possibilities to develop new approaches and applications including network characterization, comparison, classification, modeling and model evaluation, differing from using a large bag of topological measures

Analysis of a list-coloring algorithm on a random graph

conference website http://www.informatik.uni-trier.de/~ley/db/conf/focs/focs97.html ©1997 IEEE.We introduce a natural k-coloring algorithm and analyze its performance on random graphs with constant expected degree c (Gn,p = c/n). For k = 3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3-colorable. This improves the lower bound on the threshold for random 3-colorability significantly and settles the last case of a long-standing open question of Bollobas. We also provide a tight asymptotic analysis of the algorithm. We show that for all k≥3, if c≤k ln k-3/2k then the algorithm almost surely succeeds, while for any ε>0, and k sufficiently large, if c≥(1+ε)k ln k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.Technical Societies, geographic Regions, and specialized Boards/Committees

The solution space geometry of random linear equations

We consider random systems of linear equations over GF(2) in which every equation binds k variables. We obtain a precise description of the clustering of solutions in such systems. In particular, we prove that with probability that tends to 1 as the number of variables, n, grows: for every pair of solutions σ,τ, either there exists a sequence of solutions starting at σ and ending at τ such that successive solutions have Hamming distance O(log n), or every sequence of solutions starting at σ and ending at τ contains a pair of successive solutions with distance Ω(n). Furthermore, we determine precisely which pairs of solutions are in each category. Key to our results is establishing the following high probability property of cores of random hypergraphs which is of independent interest. Every vertex not in the r-core of a random k-uniform hypergraph can be removed by a sequence of O(log n) steps, where each step amounts to removing one vertex of degree strictly less than r at the time of removal. © 2013 Wiley Periodicals, Inc

The existence of uniquely -G colourable graphs

The author can archive pre-print, post-print of the article. appropriate journal homepage link is attached.Given graphs F and G and a nonnegative integer k, a function π: V(F) → {1,…,k} is a -G k-colouring of F if no induced copy of G is monochromatic; F is -G k-chromatic if F has a -G k-colouring but no -G (k − 1)-colouring. Further, we say F is uniquely -G k-colourable if F is -G k-chromatic and, up to a permutation of colours, it has only one -G k-colouring. Such notions are extensions of the well-known corresponding definitions from chromatic theory. It was conjectured that for all graphs G of order at least two and all positive integers k there exist uniquely -G k-colourable graphs. We prove the conjecture and show that, in fact, in all cases infinitely many such graphs exist

Random constraint satisfaction: A more accurate picture

http://www.springerlink.com/content/k273822u717ph566/ The original publication is available at http://www.springerlink.comIn the last few years there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Quite intriguingly, experimental results with various models for generating random CSP instances suggest that the probability of such problems having a solution exhibits a "threshold-like" behavior. In this spirit, some preliminary theoretical work has been done in analyzing these models asymptotically, i.e., as the number of variables grows. In this paper we prove that, contrary to beliefs based on experimental evidence, the models commonly used for generating random CSP instances do not have an asymptotic threshold. In particular, we prove that asymptotically almost all instances they generate are overconstrained, suffering from trivial, local inconsistencies. To complement this result we present an alternative, single-parameter model for generating random CSP instances and prove that, unlike current models, it exhibits non-triv ial asymptotic behavior. Moreover, for this new model we derive explicit bounds for the narrow region within which the probability of having a solution changes dramatically

The existence of uniquely -G colourable graphs

Given graphs F and G and a nonnegative integer k, a function n: V(F) ~ {1..... k} is a-G k-colouring of F if no induced copy of G is monochromatic; F is-G k-chromatic if F has a-G k-colouring but no-G (k- 1)-colouring. Further, we say F is uniquely-G k-colourable if F is-G k-chromatic and, up to a permutation of colours, it has only one-G k-colouring. Such notions are extensions of the well-known corresponding definitions from chromatic theory. It was conjectured that for all graphs G of order at least two and all positive integers k there exist uniquely-G k-colourable graphs, We prove the conjecture and show that, in fact, in all cases infinitely many such graphs exist