After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.

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.

In this paper, we investigate the benefit of network coding over routing for multiple independent unicast transmissions. We compare the maximum achievable throughput with network coding and that with routing only. We show that the result depends crucially on the network model. In directed networks, or in undirected networks with integral routing requirement, network coding may outperform routing. In undirected networks with fractional routing, we show that the potential for network coding to increase achievable throughput is equivalent to the potential of network coding to increase bandwidth e#ciency, both of which we conjecture to be non-existent.

The design of cost-efficient networks satisfying certain survivability

We present a branch-and-cut algorithm to solve capacitated network design problems. Given a capacitated network and point-to-point traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixed-integer programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixed-integer rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called "knapsack branching". We also report on our computational experience using real-life data.

We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1 - and L 1 -metrics in functional analysis, ffl the max-cut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of many-fermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.

Abstract. We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the max-cut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relm:ons and connectivity of the metric polytope and its relatives. In partic~dar, using its large symmetry group, we completely describe all the 13 o:bits which form the 275 840 vertices of the 21-dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the/-skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method. 1

We describe an upper-bound algorithm for multicommodity network design problems that relies on new results for approximately solving certain linear programs, and on the greedy heuristic for set-covering problems. 1 Introduction. Network design problems are mixed-integer programs that have the following broad structure. Given a graph, and a set of "demands" -- positive amounts to be routed between pairs of vertices -- capacity must be added to the edges and/or vertices of the graph, in discrete amounts, and at minimum cost, so that a feasible routing is possible. Problem of this form are increasingly important in telecommunications applications, because of the great expense inherent in maintaining and upgrading metropolitan networks. A wide variety of special cases have been studied. For example, one may be constrained to using a fixed family of paths to carry out the routing, or to using a single path for each demand, or to using integral flows. The precise manner in which capacit...

The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NP-completeness result; 3. a method of generating large classes of facets; 4. a complete characterization of 0-1 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence and adjacency relationships and diameter for small rectangular, toric and triangular boards; 7. a complete characterization of the adjacency of extreme rays, diameter, number of 2-faces and edge connectivity for rectangular toric boards.

The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NP-completeness result; 3. a method of generating large classes of facets; 4. a complete characterization of 0-1 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence, adjacency and ...