Hardness of Routing for Minimizing Superlinear Polynomial Cost in Directed Graphs.

We study the problem of routing in directed graphs with superlinear polynomial costs, which is significant for improving the energy efficiency of networks. In this problem, we are given a directed graph G(V, E) and a set of traffic demands. Routing de units of demands along an edge e will incur a cost of f(e)(delta(e)) = mu(e)(delta(e)) (alpha) with mu(e) > 0 and alpha > 1. The objective is to find integral routing paths for minimizing Sigma(e) fe(delta e). Through developing a new labeling technique and applying it to a randomized reduction, we prove an Omega((log vertical bar E vertical bar/log log vertical bar E vertical bar)(alpha) . vertical bar E vertical bar-(1/4))- hardness factor for this problem under the assumption that NP not subset of ZPTIME(n(polylog(n))).