We present a novel approach for discretely optimizing contours on the surface of a triangle mesh. This is achieved through the use of a minimum ratio cycle (MRC) algorithm, where we compute a contour having the minimal ratio between a novel contour energy term and the length of the contour. Given an initial contour, we seek to find the optimal contour within a prescribed search domain. The domain of admissible contours is modeled by a weighted acyclic edge graph, where nodes in the graph correspond to directed edges in the mesh. The acyclicity of this graph allows for an efficient computation of the MRC. To further improve the result, the algorithm may be run on a refined mesh to allow for smoother contours that can cut across mesh faces. We demonstrate the effectiveness of our algorithm in postprocessing for mesh segmentation