Abstract not found

Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs. We study two different approaches for solving problems of this type. The first one is based on the idea of metric inequalities (see Onaga and Kakusho[1971]), and uses a formulation with only jAj variables. The second uses an aggregated multicommodity flow formulation and has jV j \Delta jAj variables. We first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on 3 nodes. Next we explain our solution methods for both of the approaches in detail and present computational results. Our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real-life problems.

We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.

We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst-case improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10/9. The corresponding factor for the class of clique tree inequalities is 8/7, while it is 4/3 for the path configuration inequalities.

Abstract not found

Gomory's and Chvatal's cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is well-known that the Chvatal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1-polytope. We prove that the Chvatal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the n-dimensional 0/1-cube is at most 3n² lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n³ lg n). Moreover, we refine this result by showing that the rank of any polytope in the 0/1-cube that is defined by inequalities with small coefficients is O(n). The latter observation explains why for most cutting planes derived in polyhedral st...

Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.

Given a graph G with a specied root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to nd a minimum weight 2-hop spanning tree. The problem is NP-hard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness result for wheels and a vertex description of a linear relaxation. Some classes of valid inequalities for the convex hull of incidence vectors of 2-hop spanning trees are derived by projection techniques.

Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a xed integer L 2, the Two edge-disjoint Hop-constrained Paths Problem (THPP for short) is to nd a minimum cost subgraph such that between s and t there exist at least two edge-disjoint paths of length at most L. In this paper, we consider that problem from a polyhedral point of view. We give an integer programming formulation for the problem and discuss the associated polytope, P (G; L), when L = 2; 3. In particular, we show in this case that the linear relaxation of P (G; L), Q(G;L), given by the trivial, the st-cut, and the so-called L-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2; 3. We also give necessary and sucient conditions for these inequalities to dene facets of P (G; L) for L 2. We nally investigate the dominant of P (G; L) and give a complete description of this polyhedron for L 2, when P (G; L) = Q(G;L). Key words. Survivable network, edge-disjoint paths, hop-constraints, polyhedron, facet AMS subject classications. 90B10, 90C27, 90C57 1

In this dissertation, integer programming models are applied to combinatorial problems in air traffic flow management. For the two problems studied, models are developed and analyzed both theoretically and computationally. This dissertation makes contributions to integer programming while providing efficient tools for solving air traffic flow management problems. Currently, a constrained arrival capacity situation at an airport in the United States is alleviated by holding inbound aircraft at their departure gates. The ground holding problem (GH) decides which aircraft to hold on the ground and for how long. This dissertation examines the GH from two perspectives. First, the hubbing operations of the airlines are considered by adding side constraints to GH. These constraints enforce the desire of the airlines to temporally groupbanks of flights. Five basic models and several variations of the ground holding problem with banking constraints (GHB) are presented. A particularly strong, facet-inducing model of the banking constraints is presented which allows one to