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Computing Nonsimple Polygons of Minimum Perimeter
TU Braunschweig, Germany.
TU Braunschweig, Germany.
TU Braunschweig, Germany.
Swiss Federal Institute Technology, Switzerland.
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2016 (English)In: EXPERIMENTAL ALGORITHMS, SEA 2016, SPRINGER INT PUBLISHING AG , 2016, Vol. 9685, 134-149 p.Conference paper (Refereed)
Abstract [en]

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.

Place, publisher, year, edition, pages
SPRINGER INT PUBLISHING AG , 2016. Vol. 9685, 134-149 p.
Series
, Lecture Notes in Computer Science, ISSN 0302-9743
Keyword [en]
Traveling Salesman Problem (TSP); Minimum Perimeter Polygon (MPP); Curve reconstruction; NP-hardness; Exact optimization; Integer programming; Computational geometry meets combinatorial optimization
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-132697DOI: 10.1007/978-3-319-38851-9_10ISI: 000386324300010ISBN: 978-3-319-38850-2; 978-3-319-38851-9OAI: oai:DiVA.org:liu-132697DiVA: diva2:1048099
Conference
15th International Symposium on Experimental Algorithms (SEA)
Available from: 2016-11-20 Created: 2016-11-18 Last updated: 2016-11-20

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Schmidt, Christiane
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Communications and Transport SystemsFaculty of Science & Engineering
Discrete Mathematics

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