Theory and Methodology An optimization problem arising in the design of multiring systems Claudio Arbib * , Fabrizio Rossi Dipartimento di Matematica Pura ed Applicata, Universit a degli Studi di L'Aquila, Via Vetoio, Coppito, I-67010 L'Aquila, Italy Received 20 July 1998; accepted 7 January 1999 Abstract We address a design problem that arises when a circular architecture is adopted to move resources in a system. In many of such architectures the demand directed from one node to another aects all the intermediate nodes, involving costs which increase both with the number of nodes and with the demand volume. We consider an application to network design and propose a model for designing a minimum cost hierarchical ring network. The problem is for- malized as that of partitioning a set of nodes into p parts to be connected by rings, with the aim of minimizing the total capacity costs. We prove that the problem is NP-complete for p P 2 and propose a guaranteed approximation algo- rithm for the case p 2. We then propose a hybrid approach employing the approximation algorithm in combination with f0; 1g LP. Experiments indicate the eectiveness of this approach in terms of computational resources and solution quality. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Optimization; Hierarchical ring networks; Computational complexity; Approximation algorithms 1. Introduction In this paper we address a design problem that can arise when a circular architecture is adopted to move around resources (tools, parts, messages, and customers) in a system. The problem may indeed arise in various areas, such as telecom- munication systems, transportation or ¯exible manufacturing. We are here concerned with ap- plications in network design involving the use of ring networks as basic components. The architecture of a ring network connecting n nodes is such that these can be cyclically ordered, and a packet from the ith to the jth node has to go across nodes i k mod n, k 0; ... ; j i nd 1 i j. The problem of routing the packets, that is deciding such an order for each source± destination pair, is called ring loading and has been addressed in Refs. [1,2]. In the present paper we however assume that the routes agree for all source±destination pairs, i.e., that the ring is uni- directional. Such an assumption actually re¯ects a common practice of implementation according to which a (say) clockwise route is adopted for all the packets in normal conditions, whereas the anti- clockwise one is reserved for special use (e.g., failure). European Journal of Operational Research 124 (2000) 63±76 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +39-0862-433-138; fax: +39- 0862-433-180. E-mail address: (arbib, rossi)@univaq.it (C. Arbib). 0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 1 9 - 8