Computing Nonsimple Polygons of Minimum Perimeter
We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for a (not necessarily simply connected) polygon with shortest possible boundary length. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MPP. On the positive side, we show how to achieve a constant-factor approximation.
When trying to solve MPP instances to provable optimality by means of integer programming, an additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting additional geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that using a natural geometry-based sparsification yields results that are on average within 0.5 % of the optimum.
KeywordsTraveling Salesman Problem (TSP) Minimum Perimeter Polygon (MPP) Curve reconstruction NP-hardness Exact optimization Integer programming Computational geometry meets combinatorial optimization
We thank Stephan Friedrichs and Melanie Papenberg for helpful conversations. Parts of this work were carried out at the 30th Bellairs Winter Workshop on Computational Geometry (Barbados) in 2015. We thank the workshop participants and organizers, particularly Erik Demaine. Joseph Mitchell is partially supported by NSF (CCF-1526406). Irina Kostitsyna is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.023.208.
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