On a Routing Open Shop Problem on Two Nodes with Unit Processing Times

The Routing Open Shop Problem deals with n jobs located in the nodes of an edge-weightedgraph G=(V,E) and m machines that are initially in a special node calleddepot. The machines must process all jobs in arbitraryorder so that each machine processes at most one job at any one time and each job is processed byat most one machine at any one time. The goal is to minimize the makespan; i.e., the time whenthe last machine returns to the depot. This problem is known to be NP-hard even for the twomachines and the graph containing only two nodes. In this article we consider the particular caseof the problem with a 2-node graph, unitprocessing time of each job, and unit travel time between every two nodes. The conjecture is madethat the problem is polynomially solvable in this case; i.e., the makespan depends only on thenumber of machines and the loads of the nodes and can be calculated in time O(\log mn). We provide some new bounds on the makespan inthe case of m = n depending on the loads distribution.