An algorithm for Hierarchical Chinese postman problem using minimum spanning tree approach based on Kruskal's algorithm

Abstract

The Hierarchical Chinese postman problem is a special type of Chinese postman problem. The aim is to find a shortest tour that traverses each edge of a given graph at least once. The constraint is that the arcs are partitioned into classes and a precedence relation orders the classes according to priority. Different forms of the HCPP are applied in real life applications such as snow plowing, winter gritting and street sweeping. The problem is solvable in polynomial time if the ordering relation is linear and each class is connected. Dror et al. (1987) presented an algorithm which provides time complexity of O (kn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> ). CPP which is lower bound for HCPP. We give alternate approach by using Kruskal's method to reduce number of edges in graph which is having time complexity of O (k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), where k is number of layers in graph and n is number of nodes in graph. It is found that the suggested kruskal-based HCPP-solution gives average 21.64% improvement compare to simple HCPP and we get average 13.35% improvement over CPP when number of hierarchy is less than 3 and numbers of edges in graph are less than 10.