| 1. |
- Fekete, Sandor P., et al.
(author)
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Computing Nonsimple Polygons of Minimum Perimeter
- 2016
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In: EXPERIMENTAL ALGORITHMS, SEA 2016. - Cham : SPRINGER INT PUBLISHING AG. - 9783319388502 - 9783319388519 ; , s. 134-149
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Conference paper (peer-reviewed)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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| 2. |
- Fekete, Sándor P., et al.
(author)
-
Computing Nonsimple Polygons of Minimum Perimeter
- 2017
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In: Journal of Computational Geometry. - Ottawa, Canada : Carleton University * Department of Mathematics and Statistics. - 1920-180X. ; 8:1
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Journal article (peer-reviewed)abstract
- We consider the Minimum Perimeter Polygon Problem (MP3): for a given set V of points in the plane, find a polygon P with holes that has vertex set V , such that the total boundary length is smallest possible. The MP3 can be considered a natural geometric generalization of the Traveling Salesman Problem (TSP), which asks for a simple polygon with minimum perimeter. Just like the TSP, the MP3 occurs naturally in the context of curve reconstruction. 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 MP3. On the positive side, we provide constant-factor approximation algorithms. In addition to algorithms with theoretical worst-case guarantess, we provide practical methods for computing provably optimal solutions for relatively large instances, based on 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 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 restricting the set of connections between points to edges of the Delaunay triangulation yields results that are on average within 0.5% of the optimum for large classes of benchmark instances.
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