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The Euclidean Travelling Salesman Problem

The Euclidean Travelling Salesman Problem. Peter Eades Professor of Software Technology University of Sydney. The problem. A salesman’s territory consists of n cities. He must tour all the cities, and minimise travel time. 4571km. 4730km. Brisbane. Brisbane. Byron Bay. Byron Bay.

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The Euclidean Travelling Salesman Problem

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  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. 4571km 4730km Brisbane Brisbane Byron Bay Byron Bay Sydney Sydney Adelaide Adelaide Melbourne Melbourne • 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. • Three significant papers: • C. Papadimitriou, The Euclidean travelling salesman problem in general is NP-complete, Math. Programming 14, 312-324, 1976.

  4. Significant papers • Two significant papers: • S. Arora, "Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems", JACM 45, 753–782, 1998. • 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(n1/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.

  5. Arora’s paper Closeness to optimal solution Runtime ε

  6. Significant papers • A significant book: • D. Applegate, R. Bixby, V. Chvátal & W. Cook, The Traveling Salesman Problem: A Computational Study, Princeton University Press 2006. • This book summarises the practical breakthroughs, mainly from papers written in the 1990s. • In practice, optimal solutions for problems with about 25000 cities (eg, Sweden) can be found. • The methods used are basically variations of Integer Linear Programming.

  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.

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