Computing Nonsimple Polygons of Minimum Perimeter
2016 (English)In: EXPERIMENTAL ALGORITHMS, SEA 2016, SPRINGER INT PUBLISHING AG , 2016, Vol. 9685, 134-149 p.Conference paper (Refereed)
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.
, Lecture Notes in Computer Science, ISSN 0302-9743
Traveling Salesman Problem (TSP); Minimum Perimeter Polygon (MPP); Curve reconstruction; NP-hardness; Exact optimization; Integer programming; Computational geometry meets combinatorial optimization
IdentifiersURN: 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
15th International Symposium on Experimental Algorithms (SEA)