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Advanced Multilevel Node Separator Algorithms

  • Peter Sanders
  • Christian SchulzEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9685)

Abstract

A node separator of a graph is a subset S of the nodes such that removing S and its incident edges divides the graph into two disconnected components of about equal size. In this work, we introduce novel algorithms to find small node separators in large graphs. With focus on solution quality, we introduce novel flow-based local search algorithms which are integrated in a multilevel framework. In addition, we transfer techniques successfully used in the graph partitioning field. This includes the usage of edge ratings tailored to our problem to guide the graph coarsening algorithm as well as highly localized local search and iterated multilevel cycles to improve solution quality even further. Experiments indicate that flow-based local search algorithms on its own in a multilevel framework are already highly competitive in terms of separator quality. Adding additional local search algorithms further improves solution quality. Our strongest configuration almost always outperforms competing systems while on average computing 10 % and 62 % smaller separators than Metis and Scotch, respectively.

Keywords

Local Search Priority Queue Local Search Algorithm Breadth First Search Edge Separator 
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.

References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows: theory, algorithms, and applications (1993)Google Scholar
  2. 2.
    Bader, D., Kappes, A., Meyerhenke, H., Sanders, P., Schulz, C., Wagner, D.: Benchmarking for graph clustering and partitioning. In: Alhajj, R., Rokne, J. (eds.) Encyclopedia of Social Network Analysis and Mining. Springer, New York (2014)Google Scholar
  3. 3.
    Bhatt, S.N., Leighton, F.T.: A framework for solving vlsi graph layout problems. J. Comput. Syst. Sci. 28(2), 300–343 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bichot, C., Siarry, P. (eds.): Graph Partitioning. Wiley, New York (2011)zbMATHGoogle Scholar
  5. 5.
    Bonsma, P.: Most balanced minimum cuts. Discrete Appl. Math. 158(4), 261–276 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bui, T.N., Jones, C.: Finding good approximate vertex and edge partitions is NP-hard. Inf. Process. Lett. 42(3), 153–159 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buluç, A., Meyerhenke, H., Safro, I., Sanders, P., Schulz, C.: Recent advances in graph partitioning. In: Algorithm Engineering – Selected Topics, to app., arXiv:1311.3144 (2014)
  8. 8.
    Davis, T.: The University of Florida Sparse Matrix CollectionGoogle Scholar
  9. 9.
    Delling, D., Holzer, M., Müller, K., Schulz, F., Wagner, D.: High-performance multi-level routing. In: The Shortest Path Problem: Ninth DIMACS Implementation Challenge, vol. 74, pp. 73–92 (2009)Google Scholar
  10. 10.
    Delling, D., Sanders, P., Schultes, D., Wagner, D.: Engineering route planning algorithms. In: Lerner, J., Wagner, D., Zweig, K.A. (eds.) Algorithmics. LNCS, vol. 5515, pp. 117–139. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Dibbelt, J., Strasser, B., Wagner, D.: Customizable contraction hierarchies. In: Gudmundsson, J., Katajainen, J. (eds.) SEA 2014. LNCS, vol. 8504, pp. 271–282. Springer, Heidelberg (2014)Google Scholar
  12. 12.
    Drake, D., Hougardy, S.: A simple approximation algorithm for the weighted matching problem. Inf. Process. Lett. 85, 211–213 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fukuyama, J.: NP-completeness of the planar separator problems. J. Graph Algorithms Appl. 10(2), 317–328 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 29. WH Freeman  & Co., San Francisco (2002)Google Scholar
  15. 15.
    George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345–363 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hager, W.W., Hungerford, J.T., Safro, I.: A multilevel bilinear programming algorithm for the vertex separator problem. Technical report (2014)Google Scholar
  17. 17.
    Hamann, M., Strasser, B.: Graph bisection with pareto-optimization. In: Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments, ALENEX 2016, pp. 90–102. SIAM (2016)Google Scholar
  18. 18.
    Holtgrewe, M., Sanders, P., Schulz, C.: Engineering a scalable high quality graph partitioner. In: Proceedings of the 24th International Parallal and Distributed Processing Symposium, pp. 1–12 (2010)Google Scholar
  19. 19.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    LaSalle, D., Karypis, G.: Efficient nested dissection for multicore architectures. In: Träff, J.L., Hunold, S., Versaci, F. (eds.) Euro-Par 2015. LNCS, vol. 9233, pp. 467–478. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  21. 21.
    Leiserson, C.E.: Area-efficient graph layouts. In: 21st Symposium on Foundations of Computer Science, pp. 270–281. IEEE (1980)Google Scholar
  22. 22.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Maue, J., Sanders, P.: Engineering algorithms for approximate weighted matching. In: Demetrescu, C. (ed.) WEA 2007. LNCS, vol. 4525, pp. 242–255. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Osipov, V., Sanders, P.: n-Level graph partitioning. In: Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 278–289. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  26. 26.
    Pellegrini, F.: Scotch Home Page. http://www.labri.fr/pelegrin/scotch
  27. 27.
    Picard, J.C., Queyranne, M.: On the structure of all minimum cuts in a network and applications. Math. Program. Stud. 13, 8–16 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pothen, A., Simon, H.D., Liou, K.P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11(3), 430–452 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sanders, P., Schulz, C.: Engineering multilevel graph partitioning algorithms. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 469–480. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  30. 30.
    Sanders, P., Schulz, C.: Think locally, act globally: highly balanced graph partitioning. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 164–175. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  31. 31.
    Sanders, P., Schulz, C.: Advanced Multilevel Node Separator Algorithms. Technical report. arXiv:1509.01190 (2016)
  32. 32.
    Schloegel, K., Karypis, G., Kumar, V.: Graph partitioning for high performance scientific simulations. In: Dongarra, J., et al. (eds.) CRPC Parallel Computing Handbook. Morgan Kaufmann, San Francisco (2000)Google Scholar
  33. 33.
    C. Schulz. High Quality Graph Partititioning. Ph.D. thesis, Karlsruhe Institute of Technology (2013)Google Scholar
  34. 34.
    Schulz, F., Wagner, D., Zaroliagis, C.D.: Using multi-level graphs for timetable information in railway systems. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, pp. 43–59. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  35. 35.
    Soper, A.J., Walshaw, C., Cross, M.: A combined evolutionary search and multilevel optimisation approach to graph-partitioning. J. Global Optim. 29(2), 225–241 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Walshaw, C.: Multilevel refinement for combinatorial optimisation problems. Ann. Oper. Res. 131(1), 325–372 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany

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