Approximating Multi Commodity Network Design on Graphs of Bounded Pathwidth and Bounded Degree

In the Multicommodity Network Design problem (MCND) we are given a digraph G together with latency functions on its edges and specified flow requests between certain pairs of vertices. A flow satisfying these requests is said to be at Nash equilibrium if every path which carries a positive amount of flow is a shortest path between its source and sink. The goal of MCND is to find a subgraph H of G such that the flow at Nash equilibrium in H is optimal. While this has been shown to be hard to approximate (with multiplicative error) for a fairly large class of graphs and latency functions, we present an algorithm which computes solutions with small additive error in polynomial time, assuming the graph G is of bounded degree and bounded path-width, and the latency functions are Lipschitz-continuous. Previous hardness results in particular apply to graphs of bounded degree and graphs of bounded path-width, so it is not possible to drop one of these assumptions.