Finding the shortest route in wireless mesh networks is an important aspect. Many techniques are used to solve this problem like dynamic programming, evolutionary algorithms, weighted-sum techniques, and others. In this paper, we use dynamic programming techniques to find the shortest path in wireless mesh networks due to their generality, reduction of complexity and facilitation of numerical computation, simplicity in incorporating constraints, and their onformity to the stochastic nature of some problems. The routing problem is a multi-objective optimization problem with some constraints such as path capacity and end-to-end delay. Single-constraint routing problems and solutions using Dijkstra, Bellman-Ford, and Floyd-Warshall algorithms are proposed in this work with a discussion on the difference between them. These algorithms find the shortest route through finding the optimal rate between two nodes in the wireless networks but with bounded end-to-end delay. The Dijkstra-based algorithm is especially favorable in terms of processing time. We also present a comparison between our proposed single-constraint Dijkstra-based routing algorithm and the mesh routing algorithm (MRA) existing in the literature to clarify the merits of the former.