The Stochastic Eulerian Tour Problem

This paper defines the Stochastic Eulerian Tour Problem (SETP) and investigates several characteristics of this problem. Given an undirected Eulerian graph , a subset =) of the edges in E that require service, and a probability distribution for the number of edges in R that have to be visited in any given instance of the graph, the SETP seeks an a priori Eulerian tour of minimum expected length. We derive a closed form expression for the expected length of a given Eulerian tour when the number of required edges that have to be visited follows a binomial distribution. We also show that the SETP is NP-hard, even though the deterministic counter part is solvable in polynomial time. We derive further properties and a worst case ratio of the deviation of the expected length of a random Eulerian tour from the expected length of the optimal tour. Finally, we present some of the desirable properties in a good a priori tour using illustrative examples.