On mobile robots flow in locally uniform networks

In a mobile robot flow network different levels of flow intensity exist between the origin and destination nodes as a result of different part production requirements and mix. Henceforth, nodes handling heavy flows require additional alternative paths to the rest of the network in order to reduce blocking and traffic congestion, compared to nodes which handle moderate or light flows which require less alternative paths. In this study we make a distinction between light, moderate and heavy flows and assign one, two or three alternative node disjoint paths respectively, to handle these flows. Our aim is to find a cheapest network such that for a given node subset S of an undirected graph with a cost function on its edges, and positive integers r_i, i ∈ S, there are r_i internally node disjoint paths from each i ∈ S to any other node. We denote such a problem as the locally uniform network design problem. This problem is a special case of the network design problem and is NP-hard. The following results are obtained: For r_i ∈ {1, 2} we present an approximation algorithm that achieves a factor of 2. For r_i ∈ {1, 2, 3} we present an approximation algorithm that achieves a factor of 4 and in some cases a factor of 3.