Lecture 7 — November 23 7.1 a Fast Approximation Scheme for Maximum Mul- Ticommodity Flows

where I is the set of paths between a source-sink pair, and fI is a variable representing the flow on path I. We apply the framework of the previous lecture to obtain an algorithm for finding an ε-approximate solution of the above LP. Let A ∈ RE×I + be the constraint matrix, i.e., ae,I = 1 ce if edge e ∈ E belongs to path I ∈ I, and ae,I = 0 otherwise. Then we can rewrite the above LP as Z∗ = max{ef | Af ≤ 1, f ≥ 0}. When we attempt to apply the framework of the previous lecture, we face one problem: the number of columns corresponds to the number of paths in the graph, and hence could be exponential in general. To handle this, we keep track only of the paths with positive flows, and replace the operation of sampling from the columns by a call to a shortest path oracle. We will still maintain weights on the rows (which can be thought of as sampling probabilities), at time t, proportional to