A Laplacian Walk for the Travelling Salesman

The problem of Hamiltonian cycles is considered on the Voronoi lattice formed by N random points uniformly distributed over the unit square by using techniques borrowed from kinetic growth models. It is shown that for Laplacian random walks with appropriate boundary conditions and special growth rules one obtains nonintersecting tours covering all points of the lattice. These walks form a subset of all possible tours of the travelling salesman problem (TSP), which is well known as the canonical example of an ?-complete optimization problem. Used as a heuristic methods for the TSP, the Laplacian walk method leads to tour lengths which are comparable to Opt(2) tours.