Choosing Paths to Minimize Congestion using Randomized Rounding

To see a more complex application of Chernoff and Union Bounds, we will consider a randomized approximation algorithm for a routing problem trying to minimize congestion. We are given a (directed) graph G = (V,E), with source-sink pairs (si, ti). Each pair should be connected with a single path Pi. The congestion (or load) Le of an edge e is the number of paths Pi using e, and our goal is to minimize the maximum load of any edge maxe Le. This problem is NP-complete, since even deciding if it can be solved with maximum load 1 is the edge-disjoint paths problem. We will derive an approximation algorithm based on LP rounding. To start, we phrase the problem as an ILP with exponentially many variables. For each pair (si, ti), and each si-ti path P , we have a variable xi,P : if xi,P = 1, this means that the pair (si, ti) uses the path P to connect; otherwise, it does not. We also have one more variable, L, the maximum load of any edge. Thus, we get the following ILP: