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Although tracing linear structures in 2D images and 3D image stacks has received much attention over the years, full automation remains elusive. In this paper, we formulate the delineation problem as one of solving a Quadratic Mixed Integer Program (Q-MIP) in a graph of potential paths, which can be done optimally up to a very small tolerance. We further propose a novel approach to weighting these paths, which results in a Q-MIP solution that accurately matches the ground truth. We demonstrate that our approach outperforms a state-of-the-art technique based on the k-Minimum Spanning Tree formulation on a 2D dataset of aerial images and a 3D dataset of confocal microscopy stacks.