Computing Nonsimple Polygons of Minimum Perimeter

  • Sándor P. Fekete
  • Andreas Haas
  • Michael Hemmer
  • Michael Hoffmann
  • Irina Kostitsyna
  • Dominik Krupke
  • Florian Maurer
  • Joseph S. B. Mitchell
  • Arne Schmidt
  • Christiane Schmidt
  • Julian Troegel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9685)

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.

Keywords

Traveling Salesman Problem (TSP) Minimum Perimeter Polygon (MPP) Curve reconstruction NP-hardness Exact optimization Integer programming Computational geometry meets combinatorial optimization 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Sándor P. Fekete
    • 1
  • Andreas Haas
    • 1
  • Michael Hemmer
    • 1
  • Michael Hoffmann
    • 2
  • Irina Kostitsyna
    • 3
  • Dominik Krupke
    • 1
  • Florian Maurer
    • 1
  • Joseph S. B. Mitchell
    • 4
  • Arne Schmidt
    • 1
  • Christiane Schmidt
    • 5
  • Julian Troegel
    • 1
  1. 1.TU BraunschweigBraunschweigGermany
  2. 2.ETH ZurichZurichSwitzerland
  3. 3.TU EindhovenEindhovenThe Netherlands
  4. 4.Stony Brook UniversityStony BrookUSA
  5. 5.Linköping UniversityLinköpingSweden

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