, Volume 61, Issue 1, pp 3–35 | Cite as

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

  • Markus Bläser
  • Christian HoffmannEmail author


We consider the multivariate interlace polynomial introduced by Courcelle (Electron. J. Comb. 15(1), 2008), which generalizes several interlace polynomials defined by Arratia, Bollobás, and Sorkin (J. Comb. Theory Ser. B 92(2):199–233, 2004) and by Aigner and van der Holst (Linear Algebra Appl., 2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle, Electron. J. Comb. 15(1), 2008) employs a general logical framework and leads to an algorithm with running time f(k)⋅n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses \(2^{3k^{2}+O(k)}\cdot n\) arithmetic operations and can be efficiently implemented in parallel.


Parameterized algorithm Tree decomposition Interlace polynomial Adjacency matrix Gaussian elimination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aigner, M., van der Holst, H.: Interlace polynomials. Linear Algebra Appl. 377, 11–30 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrzejak, A.: An algorithm for the Tutte polynomials of graphs of bounded treewidth. Discrete Math. 190(1–3), 39–54 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arratia, R., Bollobás, B., Coppersmith, D., Sorkin, G.B.: Euler circuits and DNA sequencing by hybridization. Discrete Appl. Math. 104(1–3), 63–96 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. J. Comb. Theory Ser. B 92(2), 199–233 (2004) CrossRefzbMATHGoogle Scholar
  5. 5.
    Arratia, R., Bollobás, B., Sorkin, G.B.: A two-variable interlace polynomial. Combinatorica 24(4), 567–584 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Averbouch, I., Godlin, B., Makowsky, J.A.: A most general edge elimination polynomial. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds) WG. Lecture Notes in Computer Science, vol. 5344, pp. 31–42 (2008) Google Scholar
  7. 7.
    Bläser, M., Hoffmann, C.: On the complexity of the interlace polynomial. In: Albers, S., Weil, P. (eds.) 25th International Symposium on Theoretical Aspects of Computer Science (STACS 2008), pp. 97–108, Dagstuhl, Germany, 2008. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany Google Scholar
  8. 8.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bollobás, B.: Evaluations of the circuit partition polynomial. J. Comb. Theory Ser. B 85(2), 261–268 (2002) CrossRefzbMATHGoogle Scholar
  12. 12.
    Bollobás, B., Riordan, O.: A Tutte polynomial for coloured graphs. Comb. Probab. Comput. 8(1–2), 45–93 (1999) CrossRefzbMATHGoogle Scholar
  13. 13.
    Bouchet, A.: Isotropic systems. Eur. J. Comb. 8(3), 231–244 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bouchet, A.: Graphic presentations of isotropic systems. J. Comb. Theory Ser. B 45(1), 58–76 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bouchet, A.: Tutte Martin polynomials and orienting vectors of isotropic systems. Graphs Comb. 7, 235–252 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bouchet, A.: Graph polynomials derived from Tutte–Martin polynomials. Discrete Math. 302(13), 32–38 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der Mathematischen Wissenschaften/A Series of Comprehensive Studies in Mathematics, vol. 315. Springer, Berlin (1997) zbMATHGoogle Scholar
  18. 18.
    Bénard, D., Bouchet, A., Duchamp, A.: On the Martin and Tutte polynomials. Technical report, Département d’Infornmatique, Université du Maine, Le Mans, France (1997) Google Scholar
  19. 19.
    Courcelle, B.: A multivariate interlace polynomial and its computation for graphs of bounded clique-width. Electron. J. Comb. 15(1) (2008) Google Scholar
  20. 20.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Courcelle, B., Oum, S.-i.: Vertex-minors, monadic second-order logic, and a conjecture by seese. J. Comb. Theory, Ser. B 97(1), 91–126 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math. 108(1–2), 23–52 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Danielsen, L.E., Parker, M.G.: Interlace polynomials: Enumeration, unimodality, and connections to codes. Preprint (2008). arXiv:0804.2576v1
  24. 24.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  25. 25.
    Ellis-Monaghan, J.A.: New results for the Martin polynomial. J. Comb. Theory Ser. B 74(2), 326–352 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ellis-Monaghan, J.A.: Martin polynomial miscellanea. In: Proceedings of the 30th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, pp. 19–31, Boca Raton, FL, 1999 Google Scholar
  27. 27.
    Ellis-Monaghan, J.A., Sarmiento, I.: Isotropic systems and the interlace polynomial. Preprint (2006). arXiv:math/0606641v2
  28. 28.
    Ellis-Monaghan, J.A., Sarmiento, I.: Distance hereditary graphs and the interlace polynomial. Comb. Probab. Comput. 16(6), 947–973 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Intractability of clique-width parameterizations. SIAM J. Comput. 39(5), 1941–1956 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Jaeger, F.: On Tutte polynomials and cycles of plane graphs. J. Comb. Theory Ser. B 44(2), 127–146 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    JaJa, J.: Introduction to Parallel Algorithms. Addison-Wesley, Reading (1992) zbMATHGoogle Scholar
  32. 32.
    Kloks, T.: Treewidth. Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994) zbMATHGoogle Scholar
  33. 33.
    Las Vergnas, M.: Eulerian circuits of 4-valent graphs imbedded in surfaces. In: Algebraic Methods in Graph Theory, Szeged, Hungary, 1978. Colloq. Math. Soc. János Bolyai, vol. 25, pp. 451–477. North-Holland, Amsterdam (1981) Google Scholar
  34. 34.
    Las Vergnas, M.: Le polynôme de Martin d’un graphe eulerian. Ann. Discrete Math. 17, 397–411 (1983) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Las Vergnas, M.: On the evaluation at (3,3) of the Tutte polynomial of a graph. J. Comb. Theory Ser. B 45(3), 367–372 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lecerf, G., Schost, É.: Fast multivariate power series multiplication in characteristic zero. SADIO Electron. J. 5(1) (2003) Google Scholar
  37. 37.
    Leighton, F.T.: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, San Mateo (1992) zbMATHGoogle Scholar
  38. 38.
    Martin, P.: Enumérations Eulériennes dans le multigraphes et invariants de Tutte–Grothendieck. PhD thesis, Grenoble, France (1977) Google Scholar
  39. 39.
    Negami, S.: Polynomial invariants of graphs. Trans. Am. Math. Soc. 299, 601–622 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Noble, S.D.: Evaluating the Tutte polynomial for graphs of bounded tree-width. Comb. Probab. Comput. 7(3), 307–321 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Oum, S.-i.: Rank-width and vertex-minors. J. Comb. Theory Ser. B 95(1), 79–100 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Oum, S.-i., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96(4), 514–528 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Riera, C., Parker, M.G.: One and two-variable interlace polynomials: A spectral interpretation. In: Coding and Cryptography. International Workshop, WCC 2005, Bergen, Norway, March 14–18, 2005. Lecture Notes in Computer Science, vol. 3969, pp. 397–411. Springer, Berlin (2006) Google Scholar
  44. 44.
    Traldi, L.: Binary nullity, Euler circuits and interlace polynomials. Preprint (2009). arXiv:0903.4405v1
  45. 45.
    Traldi, L.: Weighted interlace polynomials. Comb. Probab. Comput. 19(1), 133–157 (2010) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fachbereich InformatikUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations