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Algorithms Based on the Treewidth of Sparse Graphs

  • Joachim Kneis
  • Daniel Mölle
  • Stefan Richter
  • Peter Rossmanith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3787)

Abstract

We prove that given a graph, one can efficiently find a set of no more than m/5.217 + 1 nodes whose removal yields a partial two-tree. As an application, we immediately get simple algorithms for several problems, including Max-Cut, Max-2-SAT and Max-2-XSAT. All of these take a record-breaking time of O *(2 m/5.217), where m is the number of clauses or edges, while only using polynomial space. Moreover, the existence of the aforementioned node sets implies an upper bound of m/5.217 + 3 on the treewidth of a graph with m edges. Letting go of polynomial space restrictions, this can be improved to a bound of m/5.769 + O(log n) on the pathwidth, leading to algorithms for the above problems that take O *(2 m/5.769) time.

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References

  1. 1.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–21 (1993)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM monographs on discrete mathematics and applications. SIAM, Philadelphia (1999)zbMATHCrossRefGoogle Scholar
  3. 3.
    Chen, J., Kanj, I.A.: Improved exact algorithms for Max-Sat. Discrete Applied Mathematics 142(1–3), 17–27 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms. Technical Report 298, Department of Informatics, University of Bergen (May 2005)Google Scholar
  5. 5.
    Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (exponential) algorithms for the dominating set problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 245–256. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Gramm, J., Hirsch, E.A., Niedermeier, R., Rossmanith, P.: New worst-case upper bounds for MAX-2-SAT with application to MAX-CUT. Discrete Applied Mathematics 130(2), 139–155 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Madsen, B.A., Rossmanith, P.: Maximum exact satisfiability: NP-completeness proofs and exact algorithms. Technical Report RS-04-19, BRICS (October 2004)Google Scholar
  8. 8.
    Scott, A., Sorkin, G.B.: Faster algorithms for Max-CUT and Max-CSP, with polynomial expected time for sparse instances. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 382–395. Springer, Heidelberg (2003)Google Scholar
  9. 9.
    Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM Journal on Discrete Mathematics 10(4), 529–550 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Woeginger, G.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Joachim Kneis
    • 1
  • Daniel Mölle
    • 1
  • Stefan Richter
    • 1
  • Peter Rossmanith
    • 1
  1. 1.Computer Science DepartmentRWTH Aachen UniversityFed. Rep. of Germany

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