Computing Nonsimple Polygons of Minimum Perimeter

Sándor P. Fekete; Andreas Haas; Michael Hemmer; Michael Hoffmann; Irina Kostitsyna; Dominik Krupke; Florian Maurer; Joseph S. B. Mitchell; Arne Schmidt; Christiane Schmidt; Julian Troegel

arXiv · 1603.07077

Computing Nonsimple Polygons of Minimum Perimeter

Sándor P. Fekete·Andreas Haas·Michael Hemmer·Michael Hoffmann·Irina Kostitsyna·Dominik Krupke·Florian Maurer·Joseph S. B. Mitchell·Arne Schmidt·Christiane Schmidt·Julian Troegel

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Abstract

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.

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Sándor P. Fekete, Andreas Haas, Michael Hemmer, Michael Hoffmann, Irina Kostitsyna, Dominik Krupke, Florian Maurer, Joseph S. B. Mitchell, Arne Schmidt, Christiane Schmidt, Julian Troegel. 2016-03-23. Computing Nonsimple Polygons of Minimum Perimeter. https://arxiv.org/abs/1603.07077

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