Local Search for the Probabilistic Traveling Salesman Problem: a proof of the incorrectness of Bertsimas’proposed 2-p-opt and 1-shift algorithms

The probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space, where each customer requires a visit only with some known probability. A solution to this problem is an a priori tour which visits all customers, and the objective is to minimize the expected length of the a priori tour over all customer subsets, assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour. This problem belongs to the class of stochastic vehicle routing problems, a class which has received increasing attention in recent years, and which is of major importance in real world applications. Several heuristics have been proposed and tested for the probabilistic traveling salesman problem, many of which were a straightforward adaptation of heuristics for the classical traveling salesman problem. In particular, two local search algorithms (2-p-opt and 1-shift) were introduced by Bertsimas and also published in this journal. Local search algorithms repeatedly perform simple modifications of a solution (moves), accepting a move only when the cost of the move is negative. In this paper we show that the expressions for the cost evaluation of 2-p-opt and 1-shift moves, as proposed by Bertsimas, are not correct. This fact has two consequences. First, results published in several papers, which used the 2-p-opt and 1-shift local search, should be re-examined because they may be wrong. Second, it poses the question whether a new fast local search operator for the probabilistic traveling salesman problem can be derived.