Computing the Moments of Costs over the Solution Space of the TSP in Polynomial Time

We give polynomial time algorithms to compute the third and fourth moments about the mean of tour costs over the solution space of the general symmetric Travelling Salesman Problem (TSP). These algorithms complement previous work on the population variance and provide a tractable method to compute the skewness and kurtosis of the probability distribution of tour costs. The methodology is generalisable to higher moments. Experimental evidence is given that suggests the skewness asymptotically approaches a limit point as the instance size is increased in several problem types.