| This paper addresses an NP-hard problem, called NTD-CB/R, whose solution is of importance to applications requiring one or more Quality of Service (QoS). Specifically, the problem calls for a network topology that meets two objectives, i.e., minimal cost and maximum bandwidth, subject to a predefined (s, t) reliability constraint. We approach the problem by converting it into one with a single objective. This is achieved via a ratio, called $bc_r$, between network bandwidth and cost to measure the goodness of each topology, and by applying Lagrange relaxation. Then we propose a dynamic programming (DP) scheme, and propose a heuristic solution, called DPCB/R, to generate each topology using all of its $n$ (s, t) paths. This paper also proposes three heuristic path orders that allow DPCB/R to generate and use only $k\le n$ paths to reduce its time complexity while producing similar results. Extensive simulations using 125 benchmark networks with various sizes show the merits of the path-orders, and effectiveness of our approach. DPCB/R is able to generate 88% optimal results for the networks. Further, its non-optimal results have a $bc_r$ ratio, bandwidth, and cost of only up to 1.56%, 0.9%, and, 2.1% off from the optimal, respectively. Further, for a grid network that contains 299 paths it uses only 1.1% of the paths while producing a topology that is only 0.92% off from optimal, with respect to $bc_r$ metric. |