# Worst Case Analyses of Nearest Neighbor Heuristic for Finding the Minimum Weight k - cycle

## Main Article Content

## Abstract

Given a weighted complete graph ( *K _{n}*,

*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

*Corresponding author: Tel.: +66 61 267 6777

E-mail: psripratak@gmail.com

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