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Computing Nonsimple Polygons of Minimum Perimeter
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- Fekete, Sandor P. (author)
- TU Braunschweig, Germany
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- Haas, Andreas (author)
- TU Braunschweig, Germany
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- Hemmer, Michael (author)
- TU Braunschweig, Germany
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- (creator_code:org_t)
- 2016-06-01
- 2016
- English.
- In: EXPERIMENTAL ALGORITHMS, SEA 2016. - Cham : SPRINGER INT PUBLISHING AG. - 9783319388502 - 9783319388519 ; , s. 134-149
- Conference paper (peer-reviewed)
Abstract
Subject headings
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- 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.
Subject headings
- NATURVETENSKAP -- Matematik -- Diskret matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Discrete Mathematics (hsv//eng)
Keyword
- Traveling Salesman Problem (TSP); Minimum Perimeter Polygon (MPP); Curve reconstruction; NP-hardness; Exact optimization; Integer programming; Computational geometry meets combinatorial optimization
Publication and Content Type
- ref (subject category)
- kon (subject category)
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- Linköping University
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