TY - GEN
T1 - Computing nonsimple polygons of minimum perimeter
AU - Fekete, Sándor P.
AU - Haas, Andreas
AU - Hemmer, Michael
AU - Hoffmann, Michael
AU - Kostitsyna, Irina
AU - Krupke, Dominik
AU - Maurer, Florian
AU - Mitchell, Joseph S.B.
AU - Schmidt, Arne
AU - Schmidt, Christiane
AU - Troegel, Julian
N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2016.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - Computational geometry meets combinatorial Optimization
KW - Curve reconstruction
KW - Exact optimization
KW - Integer programming
KW - Minimum Perimeter Polygon (MPP)
KW - NP-hardness
KW - Traveling Salesman Problem (TSP)
UR - https://www.scopus.com/pages/publications/84977559816
U2 - 10.1007/978-3-319-38851-9_10
DO - 10.1007/978-3-319-38851-9_10
M3 - Conference contribution
AN - SCOPUS:84977559816
SN - 9783319388502
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 134
EP - 149
BT - Experimental Algorithms - 15th International Symposium, SEA 2016, Proceedings
A2 - Kulikov, Alexander S.
A2 - Goldberg, Andrew V.
PB - Springer Verlag
T2 - 15th International Symposium on Experimental Algorithms, SEA 2016
Y2 - 5 June 2016 through 8 June 2016
ER -