Abstract
Given a family of m + 1 sets of n vertices in a metric space, a bi-level optimization problem to be considered in this paper asks to find a minimum cost repetitive walk with a prescribed terminal vertex. The bi-level optimization problem is inspired by an industrial application in the printed circuit board production. When a total order of the vertex sets is fixed, two consecutive vertex sets A and B give rise to a lower-level problem of finding a minimum cost Hamiltonian path that alternately visits the two sides, A and B, and has the prescribed terminal in the vertex set A, in the induced complete bipartite graph. The cost of a travel from side A to side B is multiplied by a factor β ≥ 1, called a bias, whereas the cost of a travel from B to A simply takes a metric weight. The upper-level problem asks to find a total order of the vertex sets so that the cost of the repetitive walk, that is, the total cost of the m alternating Hamiltonian paths between every two consecutive vertex sets, is minimized. In this paper, a 1.5(1+β)2/(1+β2)-approximation algorithm is presented to the bi-level optimization problem. The approximation ratio is always bounded by 3, and it also approaches 1.5 as the bias β tends to infinity.
