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We present a new branch-and-cut algorithm for the capacitated vehicle routing problem (CVRP). The algorithm uses a variety of cutting planes, including capacity, framed capacity, generalized capacity, strengthened comb, multistar, partial multistar, extended hypotour inequalities , and classical Gomory mixed-integer cuts. For each of these classes of… (More)

Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination , 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x * , ÿnds one or… (More)

The split cuts of Cook, Kannan and Schrijver are general-purpose valid inequalities for integer programming which include a variety of other well-known cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was recently cited as an open problem by Cornuéjols & Li [10].… (More)

Chvátal-Gomory and Gomory fractional cuts are well-known cutting planes for pure integer programming problems. Various methods for strengthening them are known, for example based on subadditive functions or disjunctive techniques. We present a new and surprisingly simple strengthening procedure, discuss its properties, and present some computational results.

The General Routing Problem (GRP) is the problem of finding a minimum cost route for a single vehicle, subject to the condition that the vehicle visits certain vertices and edges of a network. It contains the Rural Postman Problem, Chinese Postman Problem and Graphical Travelling Salesman Problem as special cases. We describe a cutting plane algorithm for… (More)

Valid inequalities for 0-1 knapsack polytopes often prove useful when tackling hard 0-1 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the so-called extended cover and weight inequalities can be solved… (More)

Extended Abstract Abstract Separation is of fundamental importance in cutting-plane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research eeort has been devoted to the deenition of eeective separation procedures for families of well-structured cuts. In this paper we address the separation of Chvv atal rank-1… (More)

A variety of integer programming formulations have been proposed for Vehicle Routing Problems (VRPs), including the so-called two-and three-index formulations, the set partitioning formulation, and various formulations based on extra variables representing the flow of one or more commodities. Until now, there has not been a systematic study of how these… (More)

Non-Convex Quadratic Programming with Box Constraints is a fundamental N P-hard global optimisation problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterise their extreme points and vertices,… (More)