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Other Width Metrics

  • Chapter
Fundamentals of Parameterized Complexity

Part of the book series: Texts in Computer Science ((TCS))

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Abstract

Treewidth is the best studied and perhaps the most important of a large number of width metrics in graphs. In this chapter we will explore some other metrics which have proven useful.

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Notes

  1. 1.

    As here we care about the exact parameter dependence the constants may vary depending on whether we parameterize by treewidth or branchwidth.

  2. 2.

    Alon and Gutner show that a linear time algorithm for the problem for cycles of size 6 reduces to the classical problem for finding triangles in general graphs in time O(|V|2), a well known longstanding question.

References

  1. I. Adler, Directed tree-width examples. J. Comb. Theory, Ser. B 97(5), 718–725 (2007)

    Article  MATH  Google Scholar 

  2. J. Alber, F. Dorn, R. Niedermeier, Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs. Discrete Appl. Math. 145(2), 219–231 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Alber, H. Fan, M.R. Fellows, H. Fernau, R. Niedermeier, F.A. Rosamond, U. Stege, A refined search tree technique for Dominating Set on planar graphs. J. Comput. Syst. Sci. 71(4), 385–405 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Alber, R. Niedermeier, Improved tree decomposition based algorithms for domination-like problems, in LATIN 2002: Theoretical Informatics. Proceedings of 5th Latin American Symposium, Cancun, Mexico, April 2002, ed. by S. Rajsbaum. LNCS, vol. 2286 (Springer, Berlin, 2002), pp. 613–627

    Chapter  Google Scholar 

  5. N. Alon, S. Gutner, Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. Algorithmica 54(4), 544–556 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Bader, C. Hogue, An automated method for finding molecular complexes in large protein interaction networks. BCM Bioinformatics 4 (2003)

    Google Scholar 

  7. D. Berwanger, A. Dawar, P. Hunter, S. Kreutzer, DAG-width and parity games, in Proceedings of 23rd Annual Symposium on Theoretical Aspects of Computer Science, STACS 2006, Marseille, France, February 23–25, 2006, ed. by B. Durand, W. Thomas. LNCS, vol. 3884 (Springer, Berlin, 2006), pp. 524–536

    Google Scholar 

  8. N. Betzler, R. Niedermeier, J. Uhlmann, Tree decompositions of graphs: saving memory in dynamic programming. Discrete Optim. 3(3), 220–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Fourier meets Möbius: fast subset convolution, in Proceedings of 39th ACM Symposium on Theory of Computing (STOC ’07), San Diego, California, June 11–June 13, 2007, ed. by D. Johnson, U. Feige (ACM, New York, 2007), pp. 67–74

    Google Scholar 

  10. H. Bodlaender, D. Thilikos, Constructive linear time algorithms for branchwidth, in Proceedings of 24th International Colloquium on Automata, Languages and Programming (ICALP 1997), Bologna, Italy, July 7–11, 1997, ed. by P. Degano, R. Gorrieri, A. Marchetti-Spaccamela. LNCS, vol. 1256 (Springer, Berlin, 1997), pp. 627–637

    Chapter  Google Scholar 

  11. D. Coppersmith, S. Winograd, Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Courcelle, The monadic second-order theory of graphs VII: Graphs as relational structures. Theor. Comput. Sci. 101, 3–33 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Courcelle, J. Engelfriet, Graph Structure and Monadic Second Order Logic: A Language-Theoretic Approach. Encyclopedia of Mathematics and Its Applications, vol. 138 (Cambridge University Press, Cambridge, 2012)

    Book  Google Scholar 

  14. B. Courcelle, J. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Courcelle, S. Olariu, Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–144 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. van Rooij, J.O. Wojtaszczyk, Solving connectivity problems parameterized by treewidth in single exponential time, in IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22–25, 2011, ed. by R. Ostrovsky (IEEE Comput. Soc., Los Alamitos, 2011)

    Google Scholar 

  17. F. Dorn, Dynamic programming and fast matrix multiplication, in Algorithms—ESA 2006: Proceedings of 14th Annual European Symposium, Zurich, Switzerland, September 11–13, 2006, ed. by Y. Azar, T. Erlebach. LNCS, vol. 4168 (Springer, Berlin, 2006), pp. 280–291

    Chapter  Google Scholar 

  18. F. Dorn, F. Fomin, D. Thilikos, Fast subexponential algorithm for non-local problems on graphs of bounded genus, in Algorithm Theory—SWAT 2006, Proceedings of 10th Scandinavian Workshop on Algorithm Theory, Riga, Latvia, July 6–8, 2006, ed. by L. Arge, R. Freivalds. LNCS, vol. 4059 (Springer, Berlin, 2006), pp. 172–183

    Chapter  Google Scholar 

  19. F. Dorn, F. Fomin, D. Thilikos, Catalan structures and dynamic programming in H-minor-free graphs, in Proceedings of the Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20–22, 2008, ed. by S.-H. Teng (SIAM, Philadelphia, 2008), pp. 631–640

    Google Scholar 

  20. F. Dorn, E. Penninkx, H. Bodlaender, F. Fomin, Efficient exact algorithms on planar graphs: exploiting sphere cut decompositions. Algorithmica 58(3), 790–810 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Erdös, A. Hajnal, On chromatic number of graphs and set-systems. Acta Math. Hung. 17(1–2), 61–99 (1966)

    Article  MATH  Google Scholar 

  22. M. Fellows, F. Rosamond, U. Rotics, S. Szeider, Clique-width is NP-complete. SIAM J. Discrete Math. 23(2), 909–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Fomin, P. Golovach, D. Lokshtanov, S. Saurabh, Clique-width: on the price of generality, in Proceedings of the Twentieth Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4–6, 2009, ed. by C. Mathieu (SIAM, Philadelphia, 2009), pp. 825–834

    Google Scholar 

  24. J. Fouhy, Computational experiments on graph width metrics, Master’s thesis, Victoria University of Wellington, 2002

    Google Scholar 

  25. M. Golumbic, U. Rotics, On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci. 11(3), 423–443 (2000)

    Article  MathSciNet  Google Scholar 

  26. M. Grohe, D. Marx, Descriptive complexity, canonisation, and definable graph structure theory, to appear

    Google Scholar 

  27. M. Gyssens, J. Paredaens, A decomposition methodology for cyclic databases, in Advances in Database Theory, Vol. 2, Based on the Proceedings of the Workshop on Logical Data Bases ADBT 1982, Toulouse, France (Centre d’études et de recherches de Toulouse, Toulouse, 1982), pp. 85–122

    Google Scholar 

  28. I. Hicks, Planar branch decompositions I: The ratcatcher. INFORMS J. Comput. 17(4), 402–412 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. I. Hicks, Planar branch decompositions II: The cycle method. INFORMS J. Comput. 17(4), 413–421 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Hliněný, S.-I. Oum, Finding branch-decompositions and rank-decompositions. SIAM J. Comput. 38(3), 1012–1032 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Hliněný, S.-I. Oum, D. Seese, G. Gottlob, Width parameters beyond treewidth and their applications. Comput. J. 51(3), 326–362 (2008)

    Article  Google Scholar 

  32. F. Hüffner, Algorithm engineering for optimal graph bipartization, in WEA 2005: 4th International Workshop on Efficient and Experimental Algorithms, Santorini Island, Greece, May 10–13, 2005, ed. by S. Nikoletseas. LNCS, vol. 3503 (Springer, Berlin, 2005), pp. 240–252

    Google Scholar 

  33. P. Hunter, S. Kreutzer, Digraph measures: Kelly decompositions, games and orderings, in Proceedings of the Eighteenth Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7–9, 2007, ed. by N. Bansal, K. Pruhs, C. Stein (SIAM, Philadelphia, 2007), pp. 637–644

    Google Scholar 

  34. R. Impagliazzo, R. Paturi, The complexity of k-sat, in Proceedings of the 14th Annual IEEE Conference on Computational Complexity, CCC 1999, Atlanta, GA, USA, May 4–6, 1999 (IEEE Comput. Soc., Los Alamitos, 1999), pp. 237–240

    Google Scholar 

  35. T. Johnson, N. Robertson, P. Seymour, R. Thomas, Addendum to Directed tree-width. http://www.math.gatech.edu/~thomas/PAP/diradd.pdf

  36. T. Johnson, N. Robertson, P. Seymour, R. Thomas, Directed tree-width. J. Comb. Theory, Ser. B 82(1), 138–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Kirousis, D. Thilikos, The linkage of a graph. SIAM J. Comput. 25(3), 626–647 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Kolaitis, M. Vardi, Conjunctive query containment and constraint satisfaction. J. Comput. Syst. Sci. 61(2), 302–332 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Langer, P. Rossmanith, S. Sikdar, Linear-time algorithms for graphs of bounded rankwidth: a fresh look using game theory, in CoRR (Computing Research Repository, Berlin, submitted). arXiv:1102.0908

  40. D. Lokshtanov, D. Marx, S. Saurabh, Known algorithms on graphs of bounded treewidth are probably optimal, in Proceedings of the Twenty-Second Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23–25, 2011, ed. by D. Randall (SIAM, Philadelphia, 2011), pp. 777–789

    Google Scholar 

  41. D. Lokshtanov, D. Marx, S. Saurabh, Slightly superexponential parameterized problems, in Proceedings of the Twenty-Second Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23–25, 2011, ed. by D. Randall (SIAM, Philadelphia, 2011), pp. 760–776

    Google Scholar 

  42. J. Obdržálek, Algorithmic analysis of parity games, PhD thesis, University of Edinburgh, 2007

    Google Scholar 

  43. S. Oum, P. Seymour, Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96, 514–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. G. Philip, V. Raman, S. Sikdar, Solving dominating set in larger classes of graphs: FPT algorithms and polynomial kernels, in Algorithms—ESA 2009: Proceedings of 17th Annual European Symposium, Copenhagen, Denmark, September 7–9, 2009, ed. by A. Fiat, P. Sanders. LNCS, vol. 5757 (Springer, Berlin, 2009), pp. 694–705

    Chapter  Google Scholar 

  45. G. Philip, V. Raman, S. Sikdar, Polynomial kernels for dominating sets in graphs of bounded degeneracy and beyond. ACM Trans. Algorithms 9(1), 11 (2012)

    Article  MathSciNet  Google Scholar 

  46. N. Robertson, P. Seymour, Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52(2), 153–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  47. D. Rose, R. Tarjan, Algorithmic aspects of vertex elimination on directed graphs. SIAM J. Appl. Math. 34(1), 176–197 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  48. J. Rué, I. Sau, D. Thilikos, Dynamic programming for graphs on surfaces, in Proceedings of 37th International Colloquium on Automata, Languages and Programming (ICALP 2010), Bordeaux, France, July 6–10, 2010, ed. by S. Abramsky, C. Gavoille, C. Kirchner, F.M. auf der Heide, P. Spirakis. LNCS, vol. 6198 (Springer, Berlin, 2010), pp. 372–383

    Chapter  Google Scholar 

  49. P. Seymour, R. Thomas, Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  50. G. Szekeres, H. Wilf, An inequality for the chromatic number of a graph. J. Comb. Theory, Ser. B 4, 1–3 (1968)

    Article  MathSciNet  Google Scholar 

  51. J. van Rooij, H. Bodlaender, P. Rossmanith, Dynamic programming on tree decompositions using generalised fast subset convolution, in Algorithms—ESA 2009: Proceedings of 17th Annual European Symposium, Copenhagen, Denmark, September 7–9, 2009, ed. by A. Fiat, P. Sanders. LNCS, vol. 5757 (Springer, Berlin, 2009), pp. 566–577

    Chapter  Google Scholar 

  52. V. William, Breaking the Coppersmith–Winograd barrier, to appear

    Google Scholar 

  53. G. Yuval, An algorithm for finding shortest paths using N (2.81) infinite precision multiplications. Inf. Process. Lett. 4(6), 155–156 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Statistics and Operations Research, Victoria University, Wellington, New Zealand

    Rodney G. Downey

  2. School of Engineering and Information Technology, Charles Darwin University, Darwin, Northern Territory, Australia

    Michael R. Fellows

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  1. Rodney G. Downey
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  2. Michael R. Fellows
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Downey, R.G., Fellows, M.R. (2013). Other Width Metrics. In: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-5559-1_16

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  • DOI: https://doi.org/10.1007/978-1-4471-5559-1_16

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  • Online ISBN: 978-1-4471-5559-1

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