Walking robots have many uses in manufacturing, agriculture, medical practice, prosthesis research, emergency relief, military operations, space exploration, service, and entertainment. These include industrial inspection, maintenance, personal assistance, surveillance, reconnaissance, delivery, and human interactions. In many of these scenarios, an autonomous agent must find a path through a cluttered environment to safely arrive at a destination, and such a path should in some sense be optimal. However, its workspace is generally non-convex and needs to be efficiently described in the context of an optimization problem. For this reason, we leverage the independent branching scheme to construct small, ideal formulations for constraints that ensure our robot travels only through obstacle-free polyhedral regions, which traditionally have been formulated with (non-ideal) big-M techniques. As our approach requires a biclique cover for an associated graph, we exploit the structure of this class of graphs to develop a fast subroutine for obtaining biclique covers in polynomial time. While experiments have shown the big-M approach to outperform our method on most of our test instances, there are numerous directions for future work which may improve our resolution time as well as determine classes of scenarios where our formulation may be more favorable.