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

TLDR

A single‑state machine with real variable x processes jobs that require specific start and end states, incurring a cost to change x, and this setup is a special case of the traveling salesman problem. The goal is to determine the minimal‑cost sequence for the N jobs. We present an O(N²) algorithm, and also a solution for the bottleneck variant under particular cost assumptions. The algorithm enables characterization of a special class of directed graphs that contain Hamiltonian circuits.

Abstract

We consider a machine with a single real variable x that describes its state. Jobs J 1 , …, 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(N 2 ) 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.