Testing the necklace condition for Shortest Tours and optimal factors in the plane

  • Herbert Edelsbrunner
  • Günter Rote
  • Emo Welzl
Algorithms And Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)


A tour τ of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in τ. It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour.

In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O(n2logn) time. If a tour is given, then we can test in O(n2) time whether or not this tour is a necklace tour. Both algorithms can be generalized to m-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.


Traveling salesman tour m-factor disks intersection graph transportation problem linear programming combinatorial geometry computational geometry 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
  • Günter Rote
    • 2
  • Emo Welzl
    • 3
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Institute for MathematicsTechnical University of GrazGrazAustria
  3. 3.Institute for Information ProcessingTechnical University of GrazGrazAustria

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