Edge Sparsification for Geometric Tour Problems

Abstract

We study a variety of sparsification approaches for a spectrum of geometric optimization problems related to tour problems, such as the Angular TSP, the Minimum Perimeter Problem, and the Minimum/Maximum Area Polygon Problem. To this end, we conduct a thorough study that compares the solution quality and runtime achieved by integer programming solvers on geometrically reduced edge sets to the exact solution on the full edge set; considered sparsification techniques include a variety of triangulations (Delaunay, Greedy, Minimum Weight), Theta and Yao graphs, Well-Separated Pair Decomposition, and Onion graphs.

We demonstrate that edge sparsification often leads to significantly reduced runtimes. For several of the considered problems, we can compute within a few seconds solutions that are very close to being optimal for instances that could not be solved to provable optimality within an hour; for other problems, we encounter a significant loss in solution quality. However, for almost all problems we considered, we find good solutions much earlier in the search process than for the complete edge set; thus, our methods can at least be used to provide initial bounds for the exact solution, demonstrating their usefulness even if optimality cannot be established.