Worst case analyses of nearest neighbor heuristic for finding the minimum weight k-cycle

Abstract

© 2020, King Mongkut's Institute of Technology Ladkrabang. All rights reserved. Given a weighted complete graph (Kn, w), where w is an edge weight function, the minimum weight k-cycle problem is to find a cycle of k vertices whose total weight is minimum among all k-cycles. Traveling salesman problem (TSP) is a special case of this problem when k = n. Nearest neighbor algorithm (NN) is a popular greedy heuristic for TSP that can be applied to this problem. To analyze the worst case of the NN for the minimum weight k-cycle problem, we prove that it is impossible for the NN to have an approximation ratio. An instance of the minimum weight k-cycle problem is given, in which the NN finds a k-cycle whose weight is worse than the average value of the weights of all k-cycles in that instance. Moreover, the domination number of the NN when k = n and its upper bound for the case k = n – 1 is established.