An Observation on the Repetitive Nearest Neighbour Heuristic for the Travelling Salesman Problem

Abstract

The travelling salesman problem is a well-known optimisation problem, with a large scope of application and high theoretical significance. The problem may be stated as this. A travelling salesman has to visit some cities. He starts from one city, visits every other city exactly once, going from one city to another, and comes back to the city from where he started. The cost of visiting a city from another city is given. The total cost of the visits is to be minimised. The repetitive nearest neighbour (RNN) heuristic method is a heuristic method to solve the problem. We describe a property of the RNN heuristic method of the travelling salesman problem. A general, simple and elementary example is constructed to show the property of the heuristic method. It is shown that the RNN heuristic method, too, can have efficiency that is arbitrarily bad, without any bound. This may happen even for small instances of the problem.