Quadratic TSP: A lower bounding procedure and a column generation approach

In this paper we present a Column Generation approach to the Quadratic Traveling Salesman Problem. Given a graph and a function that maps every pair of edges to a cost, the problem consists in finding a cycle that visits every vertex exactly once and such that the sum of the costs over all pairs of consecutive edges of the cycle is minimum. We propose a Linear Programming formulation that has a variable for each cycle in the graph. Since the number of cycles is exponential in the graph size, we solve our formulation via column generation. Computational results on some set of instances used in the literature show that our approach is promising. As it obtains a lower bound close to the optimal solutions for all instances.