International Journal of Pure and Applied Mathematics ————————————————————————– Volume 40 No. 3 2007, 321-340 OPTIMAL FLOW TREES FOR NETWORKS WITH GENERAL NONLINEAR ARC COSTS Dalila B.M.M. Fontes Faculdade de Economia Universidade do Porto – LIADD Rua Dr. Roberto Frias, Porto, 4200-464, PORTUGAL e-mail: fontes@fep.up.pt Abstract: This paper describes the application of a dynamic programming approach to find minimum flow cost spanning trees on a network with general nonlinear arc costs. Thus, this problem is an extension of the Minimum Span- ning Tree (MST) problem since we also consider flows that must be routed in order to satisfy user needs. In fact, the MST, usually, considers fixed arc costs and in our case the arc cost functions are nonlinear, having in addition to the fixed cost a flow dependent component. The arc cost functions involved may be of any type of form as long as they are separable and additive. This is a new problem, which is NP-Hard and a dynamic programming approach was developed to solve it exactly for small and medium size problems. We also report computational experiments on over 1200 problem instances taken from the OR-Library. AMS Subject Classification: 90C27, 90C35, 90C39 Key Words: dynamic programming, network flows, optimal trees, general nonlinear arc costs 1. Introduction In this paper, we propose to solve a new problem that consists of finding a flow tree spanning all vertices of a given network. The network consists of a central service provider designated by source vertex and a set of other vertices designated by user vertices, since they have a positive demand. In the network all vertices are connected to other vertices by a set of arcs, each of which hav- Received: July 24, 2007 c  2007, Academic Publications Ltd.