Comparing Branch-and-price Algorithms for the Unsplittable Multicommodity Flow Problem

In this paper we address branch-and-price algorithms for the unsplittable multicommodity flow problem. This problem is defined over a capacitated network in which we intend to route a set of commodities, each one with a given origin, destination and demand, at minimal cost, without exceeding the arc capacities. Furthermore, the flow of each commodity must be routed using a single path. Formulating this problem with decision variables representing flows on each arc for each commodity gives rise to a large linear (integer) program with two types of constraints: flow conservation constraints and mutual capacity constraints. Based on the Dantzig-Wolfe principle we obtain two different decompositions (path and knapsack) depending on which type of constraints defines the subproblem. In order to solve the integer problem, we combine column generation and branch-and-bound, developing branching rules that preserve the structure of the subproblem in the branch-and-bound tree for both decompositions. For the path decomposition, we compare the developed branching rule with one previously presented by Barnhart, Hane and Vance (2000). We present preliminary computational results for the two decompositions, with different branching rules, and we compare them with the ones given by a general purpose integer programming solver.