Algorithms and Complete Formulations for the Network Design Problem

We study the multi commodity uncapacitated network design problem, which is known to be NP-complete. We also study the capacitated one facility non-bifurcated version of the problem in which ow of a commodity is restricted to ow on at most two paths. We describe an exact O(n 2 2 K + n3 K) algorithm for these two versions of the multi commodity problem. This algorithm can also be used to solve the directed version of the Steiner tree problem and performs a little better than the O(n 2 2 K + n3 K + n 3) dynamic programming based algorithm due to Dreyfus and Wagner (1971). We also describe the convex hull of integer solutions to the two commodity uncapacitated problem using O(m) variables and O(n) constraints. We show that the capacitated two commodity problem can be solved either by using any standard shortest path algorithm or by the O(n 2) algorithm described for the uncapacitated case. We also describe its convex hull.