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1.5-Approximation for Treewidth of Graphs Excluding a Graph with One Crossing as a Minor

  • Erik D. Demaine
  • MohammadTaghi Hajiaghayi
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2462)

Abstract

We give polynomial-time constant-factor approximation algorithms for the treewidth and branchwidth of any H-minor-free graph for a given graph H with crossing number at most 1. The approximation factors are 1.5 for treewidth and 2.25 for branchwidth. In particular, our result directly applies to classes of nonplanar graphs such as K 5-minorfree graphs and K 3,3-minor-free graphs. Along the way, we present a polynomial-time algorithm to decompose H-minor-free graphs into planar graphs and graphs of treewidth at most c H (a constant dependent on H) using clique sums. This result has several applications in designing fully polynomial-time approximation schemes and fixed-parameter algorithms for many NP-complete problems on these graphs.

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References

  1. ABFN00.
    Jochen Alber, Hans L. Bodlaender, Henning Fernau, and Rolf Niedermeier. Fixed parameter algorithms for planar dominating set and related problems. In Algorithm theory—Scandinavian Workshop on Algorithm Theory 2000 (Bergen, 2000), pages 97–110. Springer, Berlin, 2000.Google Scholar
  2. ACP87.
    Stefan Arnborg, Derek G. Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods, 8(2):277–284, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  3. AFN01.
    Jochen Alber, Henning Fernau, and Rolf Niedermeier. Parameterized complexity: Exponential speed-up for planar graph problems. In Electronic Colloquium on Computational Complexity (ECCC). Germany, 2001.Google Scholar
  4. Asa85.
    Takao Asano. An approach to the subgraph homeomorphism problem. Theoret. Comput. Sci., 38(2–3):249–267, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  5. AST90.
    Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for graphs with excluded minor and its applications. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (Baltimore, MD, 1990), pages 293–299, 1990.Google Scholar
  6. Bak94.
    Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach., 41(1):153–180, 1994.zbMATHMathSciNetGoogle Scholar
  7. BDK00.
    Hajo J. Broersma, Elias Dahlhaus, and Ton Kloks. A linear time algorithm for minimum fill-in and treewidth for distance hereditary graphs. Discrete Appl. Math., 99(1–3):367–400, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  8. BGHK95.
    Hans L. Bodlaender, John R. Gilbert, Hjálmtýr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238–255, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  9. BKK95.
    Hans L. Bodlaender, Ton Kloks, and Dieter Kratsch. Treewidth and pathwidth of permutation graphs. SIAM J. Discrete Math., 8(4):606–616, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  10. BKMT.
    Vincent Bouchitté, Dieter Kratsch, Haiko Müller, and Ioan Todinca. On treewidth approximations. In Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW’01).Google Scholar
  11. BM76.
    John A. Bondy and U. S. R. Murty. Graph Theory with Applications. American Elsevier Publishing Co., Inc., New York, 1976.Google Scholar
  12. BM93.
    Hans L. Bodlaender and Rolf H. Möhring. The pathwidth and treewidth of cographs. SIAM J. Discrete Math., 6(2):181–188, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Bod93.
    Hans L. Bodlaender. A tourist guide through treewidth. Acta Cybernetica, 11:1–23, 1993.zbMATHMathSciNetGoogle Scholar
  14. Bod96.
    Hans L. Bodlaender. A linear-time algorithm for finding treedecompositions of small treewidth. SIAM J. Comput., 25(6):1305–1317, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  15. BT97.
    Hans L. Bodlaender and Dimitrios M. Thilikos. Treewidth for graphs with small chordality. Discrete Appl. Math., 79(1–3):45–61, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  16. BT01.
    Vincent Bouchitté and Ioan Todinca. Treewidth and minimum fill-in: grouping the minimal separators. SIAM J. Comput., 31(1):212–232 (electronic), 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  17. CKJ99.
    Jianer Chen, Iyad A. Kanj, and Weijia Jia. Vertex cover: further observations and further improvements. In Graph-theoretic concepts in computer science (Ascona, 1999), pages 313–324. Springer, Berlin, 1999.CrossRefGoogle Scholar
  18. CKL01.
    Maw-Shang Chang, Ton Kloks, and Chuan-Min Lee. Maximum clique transversals. In Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science, pages 300–310. Mathematical Programming Society, Boltenhagen, Germany, 2001.Google Scholar
  19. CNS82.
    Norishige Chiba, Takao Nishizeki, and Nobuji Saito. An approximation algorithm for the maximum independent set problem on planar graphs. SIAM J. Comput., 11(4):663–675, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  20. DF99.
    Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer-Verlag, New York, 1999.Google Scholar
  21. DHT02.
    Erik D. Demaine, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Exponential speedup of fixed parameter algorithms on K 3,3-minor-free or K 5-minor-free graphs. Technical Report MIT-LCS-TR-838, M.I.T, March 2002.Google Scholar
  22. Epp00.
    David Eppstein. Diameter and treewidth in minor-closed graph families. Algorithmica, 27(3–4):275–291, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  23. Haj01.
    MohammadTaghi Hajiaghayi. Algorithms for Graphs of (Locally) Bounded Treewidth. Master’s thesis, University of Waterloo, September 2001.Google Scholar
  24. HM94.
    Michel Habib and Rolf H. Möhring. Treewidth of cocomparability graphs and a new order-theoretic parameter. ORDER, 1:47–60, 1994.CrossRefGoogle Scholar
  25. HNRT01.
    Mohammadtaghi Hajiaghayi, Naomi Nishimura, Prabhakar Ragde, and Dimitrios M. Thilikos. Fast approximation schemes for K 3,3-minor-free or K 5-minor-free graphs. In Euroconference on Combinatorics, Graph Theory and Applications 2001 (Barcelona, 2001). 2001.Google Scholar
  26. HT73.
    J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM J. Comput., 2:135–158, 1973.CrossRefMathSciNetGoogle Scholar
  27. KC00.
    Tom Kloks and Leizhen Cai. Parameterized tractability of some (efficient) Y-domination variants for planar graphs and t-degenerate graphs. In International Computer Symposium (ICS). Taiwan, 2000.Google Scholar
  28. KLL01.
    Tom Kloks, C.M. Lee, and Jim Liu. Feedback vertex sets and disjoint cycles in planar (di)graphs. In Optimization Online. Mathematical Programming Society, Philadelphia, 2001.Google Scholar
  29. Klo93.
    Ton Kloks. Treewidth of circle graphs. In Algorithms and computation (Hong Kong, 1993), pages 108–117. Springer, Berlin, 1993.Google Scholar
  30. KM92.
    André Kézdy and Patrick McGuinness. Sequential and parallel algorithms to find a K 5 minor. In Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms (Orlando, FL, 1992), pages 345–356, 1992.Google Scholar
  31. KP02.
    Iyad A. Kanj and Ljubomir Perkovic. Improved parameterized algorithms for planar dominating set. In 27th International Symposium on Mathematical Foundations of Computer Science, MFCS 2002. Warszawa-Otwock, Poland, August 26–30, 2002. To appear.Google Scholar
  32. KR91.
    Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. J. Comput. System Sci., 42(3):288–306, 1991. Twenty-Eighth IEEE Symposium on Foundations of Computer Science (Los Angeles, CA, 1987).zbMATHCrossRefMathSciNetGoogle Scholar
  33. LT80.
    Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM J. Comput., 9(3):615–627, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  34. MR92.
    Gary L. Miller and Vijaya Ramachandran. A new graph triconnectivity algorithm and its parallelization. Combinatorica, 12(1):53–76, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  35. RS84.
    Neil Robertson and Paul D. Seymour. Graph minors. III. Planar tree-width. Journal of Combinatorial Theory Series B, 36:49–64, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  36. RS85.
    Neil Robertson and Paul D. Seymour. Graph minors — a survey. In I. Anderson, editor, Surveys in Combinatorics, pages 153–171. Cambridge Univ. Press, 1985.Google Scholar
  37. RS86.
    Neil Robertson and Paul D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms, 7(3):309–322, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  38. RS91.
    Neil Robertson and Paul D. Seymour. Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory Series B, 52:153–190, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  39. RS93.
    Neil Robertson and Paul Seymour. Excluding a graph with one crossing. In Graph structure theory (Seattle, WA, 1991), pages 669–675. Amer. Math. Soc., Providence, RI, 1993.Google Scholar
  40. SSR94.
    Ravi Sundaram, Karan S. Singh, and Pandu C. Rangan. Treewidth of circular-arc graphs. SIAM J. Discrete Math., 7(4):647–655, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  41. ST94.
    Paul D. Seymour and Robin Thomas. Call routing and the ratcatcher. Combinatorica, 14(2):217–241, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  42. Tar72.
    Robert Tarjan. Depth-first search and linear graph algorithms. SIAM J. Comput., 1(2):146–160, 1972.zbMATHCrossRefMathSciNetGoogle Scholar
  43. Wag37.
    Kehrer Wagner. Über eine Eigenschaft der eben Komplexe. Deutsche Math., 2:280–285, 1937.zbMATHGoogle Scholar
  44. Wil84.
    S. G. Williamson. Depth-first search and Kuratowski subgraphs. J. Assoc. Comput. Mach., 31(4):681–693, 1984.zbMATHMathSciNetGoogle Scholar
  45. Yan78.
    Mihalis Yannakakis. Node-and edge-deletion NP-complete problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA, 1978), pages 253–264. ACM press, New York, 1978.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • MohammadTaghi Hajiaghayi
    • 1
  • Dimitrios M. Thilikos
    • 2
  1. 1.Laboratory for Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain

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