# Linear tolls suffice: New bounds and algorithms for tolls in single source networks

## Abstract

We show that tolls that are linear in the latency of the maximum latency path are necessary and sufficient to induce heterogeneous network users to independently choose routes that lead to traffic with minimum average latency. This improves upon the earlier bound of O(n3lmax) given by Cole, Dodis, and Roughgarden in STOC 03. (Here, n is the number of nodes in the network; and lmax is the maximum latency of any edge.) Our proof is also simpler, relating the Nash flow to the optimal flow as flows rather than cuts. We model the set of users as the set [0,1] ordered by their increasing willingness to pay tolls to reduce latency - their valuation of time. Cole et al. give an algorithm that computes optimal tolls for a bounded number of agent valuations, under the very strong assumption that they know which path each user type takes in the Nash flow imposed by these (unknown) tolls. We show that in series parallel graphs, the set of paths traveled by users in any Nash flow with optimal tolls is independent of the distribution of valuations of time of the users. In particular, for any continuum of users (not restricted to a finite number of valuation classes) in series parallel graphs, we show how to compute these paths without knowing α. We give a simple example to demonstrate that if the graph is not series parallel, then the set of paths traveled by users in the Nash flow depends critically on the distribution of users' valuations of time. © 2005 Elsevier B.V. All rights reserved.