A Heuristic for the Asymmetric Traveling Salesman Problem

The traveling–salesman problem (TSP) is one of the most studied problems in combinatorial optimization [16, 14]. The TSP is simply stated, has practical applications, and is a representative of a large class of important scientific and engineering problems. The TSP can be viewed as a graph–theory problem if the cities are identified with the vertices of a graph, and the links between the cities are associated with arcs. A weight corresponding to the inter–city distance is assigned to each arc. The TSP is equivalent to finding a minimal weighted Hamiltonian circuit in the complete graph Kn. However, in its usual physical interpretation, where the vertices of a graph are cities and edges represent roads interconnecting them, the graph is most likely not complete. To remedy this situation, graph is usually completed by adding arcs with the cost of the shortest path in the original graph.