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An Optimisation Algorithm for Maximum Independent Set with Applications in Map Labelling

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Algorithms - ESA’ 99 (ESA 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1643))

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Abstract

We consider the following map labelling problem: given distinct points p 1, p 2,...,p n in the plane, find a set of pairwise disjoint axis-parallel squares Q 1,Q 2,...,Q n where p i is a corner of Q i . This problem reduces to that of finding a maximum independent set in a graph.

We present a branch and cut algorithm for finding maximum independent sets and apply it to independent set instances arising from map labelling. The algorithm uses a new technique for setting variables in the branch and bound tree that implicitly exploits the Euclidean nature of the independent set problems arising from map labelling. Computational experiments show that this technique contributes to controlling the size of the branch and bound tree. We also present a novel variant of the algorithm for generating violated odd-hole inequalities. Using our algorithm we can find provably optimal solutions for map labelling instances with up to 950 cities within modest computing time, a considerable improvement over the results reported on in the literature.

This research was (partially) supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and Complexity in Information Technology).

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Authors and Affiliations

  1. Deptartment of Computer Science, Utrecht University, The Netherlands

    Bram Verweij & Karen Aardal

Authors
  1. Bram Verweij
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  2. Karen Aardal
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Editors and Affiliations

  1. Department of Applied Mathematics and DIMATIA Centre, Charles University, Malostranská nám. 25, CZ-11800, Prague 1, Czech Republic

    Jaroslav Nešetřil

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Verweij, B., Aardal, K. (1999). An Optimisation Algorithm for Maximum Independent Set with Applications in Map Labelling. In: Nešetřil, J. (eds) Algorithms - ESA’ 99. ESA 1999. Lecture Notes in Computer Science, vol 1643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48481-7_37

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  • DOI: https://doi.org/10.1007/3-540-48481-7_37

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  • Print ISBN: 978-3-540-66251-8

  • Online ISBN: 978-3-540-48481-3

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