In the ring loading problem, traffic demands are given for each pair of nodes in an undirected ring network and a flow is routed in either of two directions, clockwise and counterclockwise. The load of an edge is the sum of the flows routed through the edge and the objective of the problem is to minimize the maximum load on the ring. Myung [J. Korean OR and MS Society, 23 (1998), pp. 49--62 (in Korean)] has presented an efficient algorithm for solving a problem where flow is restricted to integers. However, the proof for the validity of the algorithm in their paper is long and complicated and as the paper is written in Korean, its accessibility is very limited. In this paper, we slightly modify their algorithm and provide a simple proof for the correctness of the proposed algorithm.

Abstract This paper considers the Generalized Minimum Spanning Tree Problem (GMSTP). Given an undirected graph whose nodes are partitioned into mutually exclusive and exhaustive node sets, The GMSTP is then to find a minimum‐cost tree which includes exactly one node from each node set. Here, we show that the GMSTP is NP‐hard and that unless P = NP no polynomial‐time heuristic algorithm with a finite worst‐case performance ratio can exist for the GMSTP. We present various integer programming formulations for the problem and compare their linear programming relaxations. Based on the tightest formulation among the ones proposed, a dual‐based solution procedure is developed and shown to be efficient from computing experiments.

Abstract We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut relaxation in the natural space corresponding to the Steiner tree problem. © 1993 by John Wiley & Sons, Inc.