3.1 Routing to Minimize Congestion 13.1.1 Ilp Formulation with Exponential Number of Variables and Constraints

We are given a graph G = (V, E) and k " commodities ". Each commodity i has a source s i and a destination t i. The goal is to find an (s i , t i) path in the graph G for every commodity i, while minimizing the maximum congestion along any edge. The latter can be written formally as Minimize congestion C = max e |{i : P i e}|, where P i is the (s i , t i) path of commodity i and e is an edge. It can be noted that this problem is similar to a network flow problem. But it is not a network flow problem because for each commodity i, we need to have a single path between s i and t i. We cannot have the flow splitting along multiple branches. This problem is an NP-hard problem. So we will solve it by formulating it as an ILP problem, relaxing it to an LP problem, solving the LP, and rounding the solution to an ILP solution. Note that the actual objective function which is in a min-max form is not linear. So, we employ a trick of introducing a new variable t. We introduce the constraint (13.1.2) that congestion on any edge ≤ t, and so all we need to do now is minimze t, which is equivalent to minimizing the maximum congestion. The constaint 13.1.1 makes sure that we select exactly one path for each commodity. The problem with this ILP formulation is that we can have an exponential number of paths between s i and t i (for example, in a completely connected graph). So even if we relax it to an LP problem,