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Polynomial Time Approximation Schemes for MAX-BISECTION on Planar and Geometric Graphs

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STACS 2001 (STACS 2001)

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Abstract

The Max-Bisection and Min-Bisection problems are to find a partition of the vertices of a graph into two equal size subsets that respectively maximizes or minimizes the number of edges with endpoints in both subsets. We design the first polynomial time approximation scheme for the Max- Bisection problem on arbitrary planar graphs solving a long time standing open problem. The method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs, in particular Max- and Min-Bisection, which could be of independent interest. Using similar method we design also the first polynomial time approx- imation scheme for Max-Bisection on unit disk graphs (which could be easily extended to other geometrically defined graphs).

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References

  1. S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability-A survey, BIT, 25 (1985), pp. 2–23.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Arora, D. Karger and M. Karpinski. Polynomial Time Approximation Schemes for Dense Instances of NP-hard Problems, Proceedings 27th ACM Symposium on the Theory of Computing, pp. 284–293, 1995.

    Google Scholar 

  3. A.A. Ageev and M.I. Sviridenko. Approximation algorithms for Maximum Cover-age and Max Cut with cardinality constraints. Proceedings of the Conference of Integer Programming and Combinatorial Optimization 99, LNCS 1610, pp. 17–30, 1999.

    Chapter  Google Scholar 

  4. B.S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Proceedings of the 24th IEEE Foundation of Computer Science, 1983, pp. 265–273.

    Google Scholar 

  5. H.L. Bodlaender, A tourist guide through treewidth. Acta Cybernetica, 11 (1993), pp.1–23.

    MATH  MathSciNet  Google Scholar 

  6. H.L. Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing, 25 (1996), pp.1305–1317.

    Article  MATH  MathSciNet  Google Scholar 

  7. H.L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth. Avail-able at http://www.cs.ruu.nl/~hansb/index.html.

  8. H.L. Bodlaender, Personal communication, August, 2000.

    Google Scholar 

  9. H.L. Bodlaender and K. Jansen. On the complexity of the Maximum Cut problem. Nordic Journal of Computing, 7(2000), pp. 14–31, 2000.

    MATH  MathSciNet  Google Scholar 

  10. U. Feige, M. Karpinski and M. Langberg. A Note on Approximating MAX-BISECTION on Regular Graphs. ECCC(http://www.eccc.uni-trier.de/eccc/), TR00-043 (2000).

  11. U. Feige and R. Krauthgamer. A polylogarithmic approximation of the minimum bisection. To appear in Proceedings of the Foundation of Computer Science 2000.

    Google Scholar 

  12. A. Frieze and M. Jerrum. Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18, pp. 67–81, 1997.

    Google Scholar 

  13. M.X. Goemans and D.P. Williamson. Improved approximation algorithms for max-imum cut and satisfiability problems using semidefinite programming. Journal of ACM, 42, pp. 1115–1145, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 4(1975), pp. 221–225.

    Article  MATH  MathSciNet  Google Scholar 

  15. H.B. Hunt, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.S. Rosenkrantz, R.E. Stearns. NC-approximation schemes for NP-and PSPACE-hard problems for geometric graphs. Proceedings 2nd Annual European Symposium on Algorithms, (ESA), LNCS 855, pp. 468–477, Springer Verlag, June, 1994

    Google Scholar 

  16. E. Halperin and U. Zwick, Improved approximation algorithms for maximum graph bisection problems, Manuscript, 2000.

    Google Scholar 

  17. M. Jerrum, Personal communication, August, 2000.

    Google Scholar 

  18. M. Karpinski, M. Kowaluk and A. Lingas. Approximation Algorithms for Max-Bisection on Low Degree Regular Graphs and Planar Graphs. ECCC http://www.eccc.uni-trier.de/eccc/, TR00-051 (2000).

  19. S. Khanna and R. Motwani. Towards a Syntactic Characterization of PTAS. Proceedings of the 28th ACM Symposium on the Theory of Computing, 1996, pp. 329–337.

    Google Scholar 

  20. R.J. Lipton and R.E. Tarjan. A separator theorem for planar graphs. SIAM Journal of Applied Mathematics, 36 (1979), pp. 177–189.

    Article  MATH  MathSciNet  Google Scholar 

  21. N. Robertson and P.D. Seymour, Graph minors. II. Algorithmic aspects of tree-width, Journal of Algorithms, 7 (1986), pp. 309–322.

    Article  MATH  MathSciNet  Google Scholar 

  22. Y. Ye, A O.699-approximation algorithm for Max-Bisection, Submitted to Math-ematical Programming, available at http://dollar.biz.uiowa.edu/col/ye, 1999

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Author information

Authors and Affiliations

  1. Institute of Computer Science and Applied Mathematics, Christian-Albrechts-University of Kiel, 24 098, Kiel, Germany

    Klaus Jansen & Eike Seide

  2. Department of Computer Science, University of Bonn, 53117, Bonn, Germany

    Marek Karpinsk

  3. Department of Computer Science, Lund University, 22100, Lund, Sweden

    Andrzej Lingas

Authors
  1. Klaus Jansen
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  2. Marek Karpinsk
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  3. Andrzej Lingas
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  4. Eike Seide
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Editor information

Editors and Affiliations

  1. I3S & INRIA Sophia Antipolis INRIA, CNRS, 2004 Route des Lucioles, 06902, Sophia-Antipolis, France

    Afonso Ferreira

  2. Institut für Theoretische Informatik, Fakultät Informatik, TU Dresden, 01062, Dresden, Germany

    Horst Reichel

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© 2001 Springer-Verlag Berlin Heidelberg

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Jansen, K., Karpinsk, M., Lingas, A., Seide, E. (2001). Polynomial Time Approximation Schemes for MAX-BISECTION on Planar and Geometric Graphs. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_32

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  • DOI: https://doi.org/10.1007/3-540-44693-1_32

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41695-1

  • Online ISBN: 978-3-540-44693-4

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