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. 0 1995 John Wiley & Sons, Inc.
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. 0 7993 by John Wiley & Sons, Inc.
In this paper we consider the Ring Loading Problem, which arises in the design of SONET bidirectional rings. The issue of demand splitting divides the ring loading problem into the two kinds. One allows a demand to be split and routed in two different directions and the other does not. The former kind becomes a relaxation of the latter. We present an efficient exact solution procedure for the case with demand splitting, and a two-approximation algorithm for the case without demand splitting. Computational results are also shown to prove the efficiency of the proposed procedures.
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