The approximation of the Travelling Salesperson Problem is a widely studied field where even simple approaches seem to promise good results. In this thesis, we examine two different approaches to approximating the Maximum TSP. The first, the Graph Patch Heuristic, attempts to find a solution by patching a maximum-weight cycle cover into a singular cycle. The second, called the Tunnelling Algorithm, builds on the idea of approximating the Euclidean norm with a metric defined by a polytope. Despite their drastically different methodologies, we find that both approaches reduce to a similar sub-problem and yield similar approximation ratios. While the Graph Patch Heuristic seems to outperform the Tunnelling Algorithm with larger input sizes and dimensions, we find that the Tunnelling Algorithm actually solves the Max TSP in polynomial time for common norms such as L₁ and L_infty. This thesis expands on some of the nuances of both algorithms, presenting them in a self-contained manner.