Two-Machine Routing Open Shop: How Long Is the Optimal Makespan?

We consider a routing open shop problem being a natural generalization of the metric TSP and the classic open shop scheduling problem. The maximal possible ratio of the optimal makespan and the standard lower bound for the routing open shop has already been investigated in the last few years. The two-machine case is mostly covered. It is constructively proven in 2013 that the ratio mentioned above cannot be greater than 4/3, however, we do not know of any problem instance with the value of that ratio greater than 6/5. The latter ratio is achievable for a simplest case with two nodes. On the other hand, it is known that optimal makespan is at most 6/5 of the standard lower bound for at least a few special cases of the transportation network: one is with at most three nodes, and another is a tree. In this paper, we introduce an ultimate instance reduction technique, which allows reducing the general problem into a case with at most four nodes and at most six jobs. As a by-product, we propose a new polynomially solvable case of the two-machine routing open shop problem.