1 The Euclidean Travelling Salesman Problem Peter Eades Professor of Software Technology University of Sydney

2 The problem A salesman’s territory consists of n cities. He must tour all the cities, and minimise travel time. Melbourne Sydney Brisbane Adelaide Byron Bay Melbourne Sydney Brisbane Adelaide Byron Bay 4571km 4730km We want an algorithm that gives a minimum tour.

3 Significant papers  This means that it is unlikely that we there is an efficient algorithm that returns an optimal result. Rough idea: If we could solve the Euclidean Travelling Salesman problem, then we could solve a lot of other problems. But these other problems are known to have defeated many top scientists. Therefore the Euclidean travelling salesman problem is hard. 1.C. Papadimitriou, The Euclidean travelling salesman problem in general is NP-complete, Math. Programming 14, , Three significant papers:

4 Significant papers Two significant papers:  In theory, there is an efficient algorithm that returns a result that is very close to optimal.  Given ε>0, there is an algorithm that runs in time O(n 1/ε ) and returns a travelling salesman tour that has distance at most (1+ε) times the minimum possible distance.  There is an almost polynomial-time algorithm that gets an almost-optimal solution. 2.S. Arora, "Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems", JACM 45, 753–782, 1998.

5 ε Closeness to optimal solution Runtime Arora’s paper

6 Significant papers A significant book:  This book summarises the practical breakthroughs, mainly from papers written in the 1990s. In practice, optimal solutions for problems with about cities (eg, Sweden) can be found. The methods used are basically variations of Integer Linear Programming. 3.D. Applegate, R. Bixby, V. Chvátal & W. Cook, The Traveling Salesman Problem: A Computational Study, Princeton University Press 2006.

7 Where to publish? The classical venues for algorithmics are good for papers about the Euclidean Travelling Salesman Problem: Journals (ERA rank A*)  Journal of the ACM  Mathematics of Operations Research Conferences (ERA rank A)  SODA (Symposium on Discrete Algorithms)  IPCO (Integer Programming and Combinatorial Optimisation)

8 Significant groups A significant research group: Located near Rutger’s University in New Jersey. Mostly a “Virtual Organisation”. Partners are: Rutgers University, Princeton University, AT&T Labs - Research, Bell Labs, Telcordia Technologies and NEC Laboratories America. Long history of contributions to Euclidean Travelling Salesman problem, including the “Grand Challenge”. Most famous researchers in this area are members.