Worst Case Analyses of Nearest Neighbor Heuristic for Finding the Minimum Weight k - cycle
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Abstract
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.
Keywords: minimum weight k - cycle; worst case analysis; nearest neighbor heuristic
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