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Lower Bounds and Exact Algorithms for the Graph Partitioning Problem Using Multicommodity Flows

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Algorithms — ESA 2001 (ESA 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2161))

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Abstract

In this paper new and generalized lower bounds for the graph partitioning problem are presented. These bounds base on the well known lower bound of embedding a clique into the given graph with minimal congestion. This is equivalent to a multicommodity flow problem where each vertex sends a commodity of size one to every other vertex. Our new bounds use arbitrary multicommodity flow instances for the bound calculation, the critical point for the lower bound is the guaranteed cut flow of the instances. Furthermore, a branch&bound procedure basing on these bounds is presented and finally it is shown that the new bounds are also useful for lower bounds on classes of graphs, e.g. the Butterfly and Benes graph.

This work was partially supported by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT), and by the German Science Foundation (DFG) project SFB-376.

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References

  1. L. Brunetta, M. Conforti, and G. Rinaldi. A branch-and-cut algorithm for the equicut problem. Mathematical Programming, 78:243–263, 1997.

    MathSciNet  Google Scholar 

  2. C. Bornstein, A. Litman, B. Maggs, R. Sitaraman, and T. Yatzkar. On the Bisection Width and Expansion of Butterfly Networks. In Proceedings of the 1st Merged International Parallel Processing Symposium and Symposium on Parallel and Distributed Processing (IPPS/SPDP-98), pages 144–150. IEEE Computer Society, 1998.

    Google Scholar 

  3. P.-O. Fjällström. Algorithms for graph partitioning: A survey. Linköping Electronic Articles in Computer and Information Science, 1998.

    Google Scholar 

  4. L. K. Fleischer. Approximating Fractional Multicommodity Flow Independent of the Number of Commodities. SIAM Journal on Discrete Mathematics, 13(4):505–520, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. E. Ferreira, A. Martin, C. de Souza, R. Weismantel, and L. A. Wolsey. The node capacitated graph partitioning problem: a computational study. Mathematical Programming, 81:229–256, 1998.

    MathSciNet  Google Scholar 

  6. ILOG. CPLEX 7.0 Reference Manual, 2000.

    Google Scholar 

  7. E. Johnson, A. Mehrotra, and G. Nemhauser. Min-cut clustering. Mathematical Programming, 62:133–151, 1993.

    Article  MathSciNet  Google Scholar 

  8. S. E. Karisch, F. Rendl, and J. Clausen. Solving graph bisection problems with semidefinite programming. INFORMS Journal on Computing, 12(3):177–191, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  9. F. T. Leighton. Introduction to Parallel Algorithms and Architectures. Morgan Kaufman, 1992.

    Google Scholar 

  10. T. Leighton, F. Makedon, S. Plotkin, C. Stein, E. Tardos, and S. Tragoudas. Fast Approximation Algorithms for Multicommodity Flow Problems. Journal of Computer and System Sciences, 50(2):228–243, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. D. McBride. Progress made in solving the multicommodity flow problem. SIAM Journal on Optimization, 8:947–955, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  12. K. Mehlhorn and S. Näher. LEDA, a library of efficient datatypes and algorithms. Technical report, University of Saarland, 1989.

    Google Scholar 

  13. R. Preis and R. Dieckmann. The PARTY Partitioning-Library User Guide-Version 1.1. SFB 376 tr-rsfb-96-024, University of Paderborn, 1996.

    Google Scholar 

  14. C. Souza, R. Keunings, L. A. Wolsey, and O. Zone. A new approach to minimising the frontwidth in finite element calculations. Computer Methods in Applied Mechanics and Engineering, 111:323–334, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Shahrokhi and L. Szekely. On canonical concurrent flows, crossing number and graph expansion. Combinatorics, Probability and Computing, 3:523–543, 1994.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Paderborn, Fürstenallee 11, D 33102, Paderborn

    Norbert Sensen

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  1. Norbert Sensen
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science and Heinz Nixdorf Institute, Paderborn University, 33095, Paderborn, Germany

    Friedhelm Meyer auf der Heide

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© 2001 Springer-Verlag Berlin Heidelberg

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Sensen, N. (2001). Lower Bounds and Exact Algorithms for the Graph Partitioning Problem Using Multicommodity Flows. In: auf der Heide, F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44676-1_33

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  • DOI: https://doi.org/10.1007/3-540-44676-1_33

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42493-2

  • Online ISBN: 978-3-540-44676-7

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