Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems

Abstract

We present a column-generation model and branch-and-price-and-cut algorithm for origin-destination integer multicommodity flow problems. The origin-destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity (defined in this case by an origin-destination pair) may use only one path from origin to destination. Branch-and-price-and-cut is a variant of branch-and-bound, with bounds provided by solving linear programs using column-and-cut generation at nodes of the branch-and-bound tree. Because our model contains one variable for each origindestination path, for every commodity, the linear programming relaxations at nodes of the branch-and-bound tree are solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. We devise a new branching rule that allows columns to be generated efficiently at each node of the branch-and-bound tree. Then, we describe cuts (cover inequalities) that can be generated at each node of the branch-and-bound tree. These cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry. We detail the implementation of our combined columnand-cut generation method and present computational results for a set of test problems arising from telecommunications applications. We illustrate the value of our branching rule when used to find a heuristic solution and compare branch-and-price and branch-and-price-and-cut methods to find optimal solutions for highly capacitated problems.

Selected Topics in Column Generation

“…The choice of the structure on which to impose decisions is a matter of algorithmic efficiency and performance. We remark that adding cutting planes in conjunction with column generation in a branch-and-bound search is usually called branch-and-price-and-cut, see e.g., Barnhart et al (1998a); Barnhart, Hane, and Vance (2000), and Park, Kang, and Park (1996).…”

Models and algorithms for network design problems

“…Second, it is in general impossible to project out integer variables, and thus we can not apply that methodology to problems with integer routing. Dantzig-Wolfe Decomposition has been used for (ND) by Frangioni and Gendron (2010), among others, and for the closely related multi-commodity ow problem by Barnhart et al (2000), among others. Finally, Lagrangian decomposition has been used by Crainic et al (2001), among others, together with a bundle algorithm.…”

“…These cuts are able to strengthen the linear relaxation of most problems featuring a capacity constraint with binary variables. For instance, it has been used successfully for the unsplittable (see Chapter 7) multi-commodity ow problem by Barnhart et al (2000). Similarly, the study of the robust knapsack problem, where weights belong to a polyhedron W, yielded robust cover cuts which have been used successfully for the robust bandwidth packing problem (Klopfenstein and Nace).…”

k‐Splittable delay constrained routing problem: A branch‐and‐price approach

“…We propose in this section an adaptation of the Branch-and-Price method to cope with both difficulties. Branch-and-Price, as proposed early by Barnhart et al [3] is a combination of Branch-and-Bound enumeration strategy with pricing (or Column Generation). To handle the k-splittability, a path generation scheme will be applied.…”

“…To compute integer solutions, one can embed it into a branch and bound scheme. Barnhart et al [3] proposed an efficient branching for the routing problems. It is based on the concept of node of divergence over the aggregated flow x h a = p δ p a x h p , ∀a ∈ A.…”