The centred travelling salesman at high temperature

A recently formulated statistical mechanics method is used to study the phase transition occurring in a generalisation of the Traveling Salesman Problem (TSP) known as the centred TSP. The method shows that the problem has clear signs of a crossover, but is only able to access (unscaled) finite temperatures above the transition point. The solution of the problem using this method displays a curious duality

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