Journal of Combinatorial Optimization, 4, 415–436, 2000 c  2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Generalized Steiner Problems and Other Variants MOSHE DROR MIS, College of Business and Public Administration, The University of Arizona, Tucson, Arizona, USA MOHAMED HAOUARI Ecole Polytechnique de Tunisie, BP 743, 2078 La Marsa, Tunisia Received October 29, 1998; Revised March 10, 2000; Accepted March 29, 2000 Abstract. In this paper, we examine combinatorial optimization problems by considering the case where the set N (the ground set of elements) is expressed as a union of a finite number of m nonempty distinct subsets N 1 ,..., N m . The term we use is the generalized Steiner problems coined after the Generalized Traveling Salesman Problem. We have collected a short list of classical combinatorial optimization problems and we have recast each of these problems in this broader framework in an attempt to identify a linkage between these “generalized” problems. In the literature one finds generalized problems such as the Generalized Minimum Spanning Tree (GMST), Generalized Traveling Salesman Problem (GTSP) and Subset Bin-packing (SBP). Casting these problems into the new problem setting has important implications in terms of the time effort required to compute an optimal solution or a “good” solution to a problem. We examine questions like “is the GTSP “harder” than the TSP?” for a number of paradigmatic problems starting with “easy” problems such as the Minimal Spanning Tree, Assignment Problem, Chinese Postman, Two-machine Flow Shop, and followed by “hard” problems such as the Bin-packing, and the TSP. Keywords: complexity, NP-hardness, generalized TSP, approximation algorithms 1. Introduction Perhaps one of the most dynamic areas of modern applied mathematics is combinatorial optimization. A major reason for this rapid growth is its wealth of pertinence to a wide range of applied areas including allocation of scarce resources, strategic planning, operations scheduling, transportation, network design, computer science, and even molecular biology and high-energy physics, to quote just a few. Formally, a combinatorial optimization problem may be defined as follows (Nemhauser and Wolsey, 1988): Let N ={1,..., n} be a finite set and let c ={c 1 ,..., c n } be an n-vector. For F ⊆ N define c( F ) = ∑ j ∈F c j . Suppose we are given a collection of subsets  of N . The combinatorial optimization problem is: (CO) Minimize{c( F ) : F ∈ } In this paper, we examine combinatorial optimization problems for which we require that the collection of subsets  conform to certain problem specific requirements with respect to the m ≥ 1 nonempty distinct subsets N 1 ,..., N m , given that the set N is expressed