Generalized Spanning Trees and Extensions Corinne Feremans,
[email protected] Director: Professeur Martine Labb´e,
[email protected] Codirector: Professeur Gilbert Laporte,
[email protected]
Keywords: spanning trees, integer linear programming, network design, polyhedral analysis.
1.
Abstract
This abstract is submitted in support of the INFORMS Telecommunications Dissertation Award. In this dissertation, network design problems where a global network interconnects at minimum cost local networks together (telecommunication, transportation networks) are considered. The generalized minimum spanning tree problem is defined on a graph where the vertex set is partitioned into clusters and a nonnegative cost is associated to each edge. It consists of finding a tree with minimum cost that includes exactly one vertex from each cluster. Several linear formulations are compared regarding their linear relaxation in order to determine a convenient formulation used in an exact algorithm. Several families of valid inequalities for this problem are shown to be facet-defining for the associated polytope. We have developed a Branchand-Cut algorithm that uses separation methods for these new classes of valid inequalities. A tabu heuristic is implemented providing an initial upper bound to the exact method. This exact method is tested on several types of instances: real world instances and randomly generated instances. Two extensions of the generalized minimum spanning tree problem are considered. The variant where the tree of minimum cost is allowed to contain at least one vertex in each cluster is studied. The problem of the maximum cost subgraph containing at most one vertex from each cluster is investigated.
2.
Introduction
The network is the computer. This is the slogan used to promote a famous brand of computers. It gives a taste of the aims of computer industry those days. The network has to be economical, reliable, quick and wide. New local networks arise every day and they are powerful and efficient only if they can be interconnected in an effective way. Local networks are part of our daily life: phone, energy, computer, transport networks... In a local network, several possible locations can be determined to be used as an entrance gate to plug in a new global network. In the design of a minimum cost global network interconnecting local networks together, two simultaneous decisions appear: the choice among possible locations in each local network for an entrance gate to the global network, and the design of the global network linking the selected gates. This problem occurs, for instance, when a new telecommunication operator wants to build new telecommunication links that interconnect existing networks (long distance provider that buys connection rights to local operators). In a first estimate, the cost of a connection center (connection right) from a local competitor is supposed to be so high that the new competitor is restricted to buy exactly one connection center from each local competitor. This leads to consider the Generalized Minimum Spanning Tree Problem in its “exactly” version. Since trees minimize the total cost while connecting all vertices, this problem consists of determining a tree with minimal total cost (sum of the cost of the selected links) that includes exactly one center from each local network. This problem is denoted by E-GMSTP. It was introduced by Myung, Lee and Tcha [2]. It can be formulated as a problem defined on a graph where the vertex set is partitioned into clusters. The vertices of the graph represent the possible locations for an entrance gate for all the local networks. With each local network is associated a cluster. The problem remains then to determine a minimum cost tree that includes
exactly one vertex from each cluster. In Chapter 1, we set up the framework of the generalized design network problems considered in this dissertation. We are interested in combinatorial optimization problems defined on graphs where a partition of the vertices is given and where the aim is to find a subgraph with a given structure that includes exactly one vertex from each cluster of the partition. We present a non-exhaustive list of problems defined on graphs that have a generalized structure. The purpose of chapter 2 is to analyze and compare integer linear programming formulations for the EGMSTP. Additional formulations to those introduced by Myung et al.[2] are proposed and a comparative analysis of the polytopes associated with the linear relaxations of all these formulations is provided. This analysis suggests which formulations should be preferred for the construction of an exact algorithm and for the evaluation of heuristics. Chapter 3 is dedicated to a polyhedral analysis of the E-GMSTP and of two of its relaxations. Polytopes associated with the Generalized Subgraph and the Generalized Acyclic subgraph structures are considered. A generalized subgraph (GS) is a subgraph that contains at most one vertex in each cluster. A generalized acyclic subgraph is a generalized subgraph without cycle. The GS structure subsumes several network design structures including the Generalized Travelling Salesman Problem. In Chapter 4, we further study the Generalized Subgraph Problem (GSP). This problem consists of finding a maximum cost (sum of the edge cost plus sum of the vertex cost) subgraph that contains at most one vertex in each cluster. The problem is examined in terms of complexity, approximability and polyhedral theory. Interesting connections between the GSP and several other well-known combinatorial optimization problems, including the matching, max-flow / min-cut, stable set and quadratic semi-assignment problems are emphasized. It turns out that the GSP is intermediate between the matching problem and the stable set problem. Chapter 5 describes the Branch-and-Cut (B&C) method implemented to solve the E-GMSTP. This algorithm is based on one of the best formulations from Chapter 2 and uses valid inequalities from Chapter 3 and from Chapter 4. Three tabu heuristics are developed in order to give an initial upper bound to the exact method and to produce an approximation of the solution for large instances. Chapter 6 shows computational results obtained by the exact algorithm for the E-GMSTP described in Chapter 5. Real world instances from TSPLIB [3] and randomly generated instances are tested and compared to the results obtained in Myung et al. [2]. The tested instances have up to 200 vertices and 20000 edges. As far as we know, it is the first time that one solves instances of the E-GMSTP of more than 100 vertices to optimality. Chapter 7 presents a variant of the E-GMSTP. This problem is to determine a minimum cost tree that includes at least one vertex in each cluster. This problem is denoted by L-GMSTP. A direct exact method similar to the B&C described in Chapter 5 is developed to solve the L-GMSTP. Two transformations from a L-GMSTP instance to a E-GMSTP instance are proposed. These methods are compared to heuristic methods proposed in Dror, Haouari, Chaouachi [1] on the instances provided by these authors.
References [1] Dror M., Haouari M., Chaouachi J., “Generalized Spanning Trees,” European Journal of Operational Research 120, pp. 583–592, 2000. [2] Myung Y. S., Lee C. H., Tcha D. W., “On the Generalized Minimum Spanning Tree Problem,” Networks 26, pp. 231–241, 1995. [3] Reinelt G., “TSPLIB - A Traveling Salesman Problem Library,” ORSA Journal on Computing 3, pp. 376–384, 1991. 2
