Diffusive behavior of a greedy traveling salesman

Abstract

Using Monte Carlo simulations we examine the diffusive properties of the greedy algorithm in the d-dimensional traveling salesman problem. Our results show that for d = 3 and 4 the average squared distance from the origin r 2 is proportional to the number of steps t. In the d = 2 case such a scaling is modified with some logarithmic corrections, which might suggest that d = 2 is the critical dimension of the problem. The distribution of lengths also shows marked differences between d = 2 and d > 2 versions. A simple strategy adopted by the salesman might resemble strategies chosen by some foraging and hunting animals, for which anomalous diffusive behavior has recently been reported and interpreted in terms of Lévy flights. Our results suggest that broad and Lévy-like distributions in such systems might appear due to dimension-dependent properties of a search space.