A Travelling Salesman Problem special case: 30-year-old problem solved

The Travelling Salesman Problem, or TSP, was first defined around 150 years ago. The problem then was to find the shortest possible route for salesmen to visit each of their customers once and finish back where they started. In the 21^st century, this same problem now applies to a multitude of activities -- delivering fresh stock to supermarkets, supplying manufacturing lines, air traffic control, and even DNA sequencing. Complex and sophisticated computer programmes using optimisation -- where algorithms produce the best possible result from multiple choices -- now form the basis of solutions to these modern-day problems. The time required to find an optimal solution is vital for practical application of the TSP. How long can lorry drivers wait for their route to be finalised when the salads they hope to deliver will only be fresh for another 24 hours? How long can air traffic control keep an airliner flying in circles around Heathrow Airport?

The theoretical background behind these types of questions is studied in the theory of computational complexity. The TSP is of paramount significance for this branch of knowledge. Even a small incremental step in understanding the nature of this problem is of interest and benefit to the scientific community.

Associate Professor Dr Vladimir Deineko of Warwick Business School, together with Eranda Cela (University of Technology Graz, Austria) and Gerhard Woeginger (Eindhoven University, the Netherlands) have addressed a special case of the TSP, or open problem as it is termed, first identified 30 years ago. Dr Deineko's and his colleagues' work gives a solution of theoretical significance for computer science and operational research.

Dr Deineko comments, "The TSP has served as a benchmark problem for all new and significant approaches developed in optimisation. It belongs to the set of so called NP-hard problems. There are obviously some special cases when the TSP can be solved efficiently. The simplest possible case is when the cities are the points on a straight line and the distances are as-the-crow-flies-distances. One can easily get the shortest route for visiting all the points in this case. Probably the next simplest case would be the case when the cities are the points on two perpendicular lines and the distances are again as-the-crow-flies-distances (so-called X-and-Y-axes TSP).

"Despite its apparent simplicity, this special case problem has been circulating in the scientific community for around 30 years. Until our work, it was not known whether an algorithm existed which would guarantee finding an optimal solution to any instance of these problems within a reasonable amount of time. We have now proved that the X-and-Y axes TSP can be easily solved in a number of steps proportional to the square of the number of cities."

Dean of WBS, Professor Mark Taylor, comments, "I congratulate Dr Deineko and his colleagues in advancing our knowledge of this enormously complex subject. They have produced cutting-edge research which is not only of great importance to the scientific community, but ultimately also of great relevance to all of us who depend on modern technology as we go about our daily lives."