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An important goal of bandwidth allocation is to maximize the utilization of network resources while sharing the resources in a fair manner among network flows. To strike a balance between fairness and throughput, a widely studied criterion in the network community is the notion of max-min fairness. However, the majority of work on max-min fairness has been limited to the case where the routing of flows has already been defined and this routing is usually based on a single fixed routing path for each flow. In this paper, we consider the more general problem in which the routing of flows, possibly over multiple paths per flow, is an optimization parameter in the bandwidth allocation problem. Our goal is to determine a routing assignment for each flow so that the bandwidth allocation achieves optimal utility max-min fairness with respect to all feasible routings of flows. We present evaluations of our proposed multi-path utility max-min fair allocation algorithms on a statistical traffic engineering application to show that significantly higher minimum utility can be achieved when multi-path routing is considered simultaneously with bandwidth allocation under utility max-min fairness, and this higher minimum utility corresponds to significant application performance improvements.