ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2012, Vol. 6, No. 3, pp. 318–331. c  Pleiades Publishing, Ltd., 2012. Original Russian Text c  A.V. Kononov, 2012, published in Diskretnyi Analiz i Issledovanie Operatsii, 2012, Vol. 19, No. 2, pp. 55–75. On the Routing Open Shop Problem with Two Machines on a Two-Vertex Network A. V. Kononov * Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia Received June 9, 2011; in final form, November 24, 2011 Abstract—The open shop problem is under study with two machines and routing on a two-vertex network. This problem is NP-hard. We introduce an exact pseudopolynomial algorithm, propose a fully polynomial time approximation scheme for solving this problem, and consider some particular polynomially solvable cases. DOI: 10.1134/S1990478912030064 Keywords: open shop problem, routing, fully polynomial time approximation scheme INTRODUCTION Statement of the Problem. We consider the following scheduling problem: A set of jobs N has to be processed with two machines A and B. For each job J j from N , times a j and b j are given of its processing by machines A and B correspondingly. The network consists of two vertices V 0 and V 1 connected with an edge. The set of jobs is split into the subsets N 0 and N 1 : all jobs from N 0 lie at the vertex V 0 , and all jobs from N 1 , at V 1 . In order to process each job a machine has to move into the vertex to which this job belongs. For each machine, it takes time τ to move from one vertex to another. At the initial moment, both machines are located at V 0 and have to return to this vertex after processing all jobs. Thus, each machine has to make an even number of movements between the vertices. Each job consists of exactly two operations, one of them should be executed by the machine A and the other, by B. No pair of operations executed by the same machine or belonging to the same job can be processed simultaneously. Any preemptions in operation execution are not allowed. The problem consists in finding a schedule for the execution of jobs and movement of the machines such that the machines execute all jobs and return to the initial vertex V 0 in a minimum time. We denote this by Problem ROS 2,2 . Overview of the available results. Problem ROS 2,2 is a particular case of the Routing Open Shop problem (Problem ROS), where the number of machines and the configuration of the network are arbitrary. In turn, the routing open shop problem is a generalization of the two classical discrete optimization problems: the open shop problem and the metric travelling salesman Problem. Problem ROS was for the first time considered by I. Averbakh, O. Berman, and I. Chernykh [2, 3]. Problem ROS is NP-hard in strong sense since the metric travelling salesman problem is its particular case. Moreover, it is NP-hard in the usual sense even on the two-vertex network with two machines [3]; i.e., Problem ROS 2,2 also is NP-hard. In the latter case, a 6/5-approximation algorithm is proposed in [2]. In the case with two machines and the jobs located on an arbitrary network, a 7/4-approximation algorithm is proposed in [3]; and, in the case with an arbitrary number m of machines, an (m + 4)/2-approximation algorithm is proposed. In [4], the two latter results were improved: in the case with two machines on an arbitrary network, a 13/8-approximation algorithm was constructed and, in the general case, an O( √ m)-approximation algorithm. In this paper, we consider new polynomially solvable classes of Problem ROS 2,2 and propose an exact pseudopolynomial algorithm and a fully polynomial time approximation scheme. In Section 1, we * E-mail: alvenko@math.nsc.ru 318