Sequencing a One State-Variable Machine: A Solvable Case of the Traveling Salesman Problem

We consider a machine with a single real variable x that describes its state. Jobs J₁, …, J_N are to be sequenced on the machine. Each job requires a starting state A, and leaves a final state B_i. This means that J_i can be started only when x = A_i and, at the completion of the job, x = B_i. There is a cost, which may represent time or money, etc., for changing the machine state x so that the next job may start. The problem is to find the minimal cost sequence for the N jobs. This problem is a special case of the traveling salesman problem. We give a solution requiring only 0(N²) simple steps. A solution is also provided for the bottleneck form of this traveling salesman problem under special cost assumptions. This solution permits a characterization of those directed graphs of a special class which possess Hamiltonian circuits.