ESA 1999: Algorithms - ESA’ 99 pp 426-437 | Cite as

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

  • Bram Verweij
  • Karen Aardal
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1643)

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.

Keywords

Maximum Clique Valid Inequality Path Decomposition Maximum Clique Problem Closed Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. K. Agarwal, M. van Kreveld, and S. Suri. Label placement by maximum independent set in rectangles. Comp. Geom. Theory and Appl., 11:209–218, 1998.MATHGoogle Scholar
  2. 2.
    L. Babel. Finding maximum cliques in arbitrary and special graphs. Computing, 46(4):321–341, 1991.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    L. Babel and G. Tinhofer. A branch and bound algorithm for the maximum clique problem. ZOR-Methods and Models of Opns. Res., 34:207–217, 1990.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    V. Bafna, B. O. Narayanan, and R. Ravi. Non-overlapping local alignments (weighted independent sets of axis parallel rectangles). In Selim G. Akl, Frank K. H. A. Dehne, Jörg-Rüdiger Sack, and Nicola Santoro, editors, Proc. 4th Int. Workshop on Alg. and Data Structures, volume 955 of Lec. Notes Comp. Sci., pages 506–517. Springer-Verlag, 1995.Google Scholar
  5. 5.
    E. Balas and J. Xue. Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica, 15:397–412, 1996.MATHMathSciNetGoogle Scholar
  6. 6.
    E. Balas and C. S. Yu. Finding a maximum clique in an arbitrary graph. SIAM J. on Comp., 15:1054–1068, 1986.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. Carraghan and P. M. Pardalos. An exact algorithm for the maximum clique problem. Opnl. Res. Letters, 9:375–382, 1990.MATHCrossRefGoogle Scholar
  8. 8.
    J. Christensen, J. Marks, and S. Shieber. An empirical study of algorithms for point-feature label placement. ACM Trans. on Graphics, 14(3):203–232, 1995.CrossRefGoogle Scholar
  9. 9.
    R. G. Cromley. An LP relaxation procedure for annotating point features using interactive graphics. In Proc. AUTO-CARTO VII, pages 127–132, 1985.Google Scholar
  10. 10.
    S. van Dijk, D. Thierens, and M. de Berg. On the design of genetic algorithms for geographical applications. In W. Banzhaf, J. Daida, A. E. Eiben, M. H. Garzon, V. Honavar, M. Jakiela, and R. E. Smith, editors, Proc. of the Genetic and Evolutionary Comp. Conf. (GECCO-99). Morgan Kaufmann, Jul 1999.Google Scholar
  11. 11.
    E. Dijkstra. A note on two problems in connexion with graphs. Numeriche Mathematics, 1:269–271, 1959.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    B. de Fluiter. Algorithms for Graphs of Small Treewidth. PhD thesis, Utrecht University, Utrecht, 1997.MATHGoogle Scholar
  13. 13.
    M. Formann and F. Wagner. A packing problem with applications to lettering of maps. In Proc. 7th Ann. ACM Symp. Comp. Geom., pages 281–288, 1991.Google Scholar
  14. 14.
    C. Friden, A. Hertz, and D. deWerra. An exact algorithm based on tabu search for finding a maximum independent set in graph. Comp. Opns. Res., 17(5):375–382, 1990.Google Scholar
  15. 15.
    M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, 1988.MATHGoogle Scholar
  16. 16.
    ILOG, Inc. CPLEX Devision. Using the CPLEX Callable Library, 1997.Google Scholar
  17. 17.
    K. G. Kakoulis and I. G. Tollis. A unified approach to labeling graphical features. In Proc. 14th Ann. ACM Symp. Comp. Geom., pages 347–356, 1998.Google Scholar
  18. 18.
    L. Kučera, K. Mehlhorn, B. Preis, and E. Schwarzenecker. Exact algorithms for a geometric packing problem. In Proc. 10th Symp. Theo. Aspects Comp. Sci., volume 665 of Lec. Notes Comp. Sci., pages 317–322. Springer-Verlag, 1993.Google Scholar
  19. 19.
    J. T. Linderoth and M. W. P. Savelsbergh. A computational study of search strategies for mixed integer programming. INFORMS J. on Comp., to appear.Google Scholar
  20. 20.
    C. Mannino and A. Sassano. An exact algorithm for the maximum stable set problem. Comp. Optn. and Appl., 3:243–258, 1994.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    K. Mehlhorn, S. Näher, and C. Uhrig. The LEDA User Manual Version 3.7. Max-Planck-Institut für Informatik, Saarbrücken, 1998.Google Scholar
  22. 22.
    G. L. Nemhauser and G. Sigismondi. A strong cutting plane/branch-and-bound algorithm for node packing. J. of the Opns. Res. Soc., 43(5):443–457, 1992.MATHGoogle Scholar
  23. 23.
    M. W. Padberg. On the facial structure of set packing polyhedra. Mathematical Programming, 5:199–215, 1973.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    F. Rossi and S. Smriglio. A branch-and-cut algorithm for the maximum cardinality stable set problem. Technical Report 353, “Centro Vito Volterra”-Università di Roma Tor Vergata, Jan 1999.Google Scholar
  25. 25.
    E. C. Sewell. A branch and bound algorithm for the stability number of a sparse graph. INFORMS J. on Comp., 10(4):438–447, 1998.MathSciNetCrossRefGoogle Scholar
  26. 26.
    F. Wagner and A. Wolff. An efficient and effective approximation algorithm for the map labeling problem. In P. Spirakis, editor, Proc. 3rd Ann. Eur. Symp. on Alg., volume 979 of Lec. Notes Comp. Sci., pages 420–433. Springer-Verlag, 1995.Google Scholar
  27. 27.
    L. Wolsey. Integer Programming. JohnWiley & Sons, Inc., NewYork, 1998.MATHGoogle Scholar
  28. 28.
    J. Xue. Fast algorithms for the vertex packing problem. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, 1991.Google Scholar
  29. 29.
    S. Zoraster. Integer programming applied to the map label placement problem. Cartographica, 23(3):16–27, 1986.Google Scholar
  30. 30.
    S. Zoraster. The solution of large 0-1 integer programming problems encountered in automated cartography. Opns. Res., 38(5):752–759, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Bram Verweij
    • 1
  • Karen Aardal
    • 1
  1. 1.Deptartment of Computer ScienceUtrecht UniversityThe Netherlands

Personalised recommendations