Computing the tutte polynomial in vertex-exponential time

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2008, Pages 677-686

  Björklund, Andreas ^a   Husfeldt, Thore ^a,b   Kaski, Petteri ^c   Koivisto, Mikko ^c  

^a LUND UNIVERSITY   (Sweden)

^b IT UNIVERSITY OF COPENHAGEN   (Denmark)

^c UNIVERSITY OF HELSINKI   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL GEOMETRY; COMPUTERS; FIBER OPTIC CHEMICAL SENSORS; GRAPH THEORY; POLYNOMIAL APPROXIMATION; POLYNOMIALS; RELIABILITY THEORY; SHRINKAGE; TOPOLOGY;

ARBITRARY GRAPHS; EXPONENTIAL TIMES; GRAPH INVARIANTS; INPUT GRAPHS; JONES POLYNOMIALS; KNOT THEORIES; PARTITION FUNCTIONS; POLYNOMIAL FACTORS; RELIABILITY POLYNOMIALS; RUNNING IN; SPANNING TREES; STATISTICAL PHYSICS; TUTTE POLYNOMIALS; VERTEX SETS;

TREES (MATHEMATICS);

EID: 57949113843     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2008.40     Document Type: Conference Paper

References (47)

1. A. Andrzejak, An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998), 39-54.
2. J. D. Annan, The complexities of the coefficients of the Tutte polynomial, Discrete Appl. Math. 57 (1995), 93-103.
3. M. H. G. Anthony, Computing chromatic polynomials, Ars Combinatoria 29 (1990), 216-220.
4. N. Biggs, Algebraic Graph Theory, 2nd ed., Cambridge University Press, 1993.
5. A. Björklund, T. Husfeldt, Exact algorithms for exact satisfiability and number of perfect matchings, Algorithmica, 2007, doi:10.1007/s00453-007-9149-8.
6. A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, Fourier meets Möbius: fast subset convolution, Proceedings of the 39th Annual ACM Symposium on Theory of Computing (San Diego, CA, June .11-13, 2007), Association for Computing Machinery, 2007, pp. 67-74.
7. A. Björklund, T. Husfeldt, P. Kaski, M. Koivisto, The travelling salesman problem in bounded degree graphs, Proceedings of the 35th International Colloquium, on Automata, Languages and Programming (Reykjavik, Iceland, July 6-13, 2008), Lecture Notes in Computer Science 5125, Springer, 2008, pp. 198-209.
8. A. Björklund, T. Husfeldt, M. Koivisto, Set partitioning via inclusion-exclusion, SIAM J. Computing, to appear.
9. M. Bläser, H. Dell, Complexity of the cover polynomial, Proceedings of the 34th International Colloquium on Automata, Languages and Programming (Wroclaw, Poland, July 9-13, 2007), Lecture Notes in Computer Science 4596, Springer, 2007, pp. 801-812.
10. B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, 1998.
11. J. A. Buzacott, A recursive algorithm for finding reliability measures related to the connection of nodes in a graph, Networks 10 (1980), 311-327.
12. A. Cayley, A theorem on trees, Quart. J. Math. 23 (1889), 376-378.
13. F. R. K. Chung, R. L. Graham, On the cover polynomial of a digraph, J. Combin. Theory Ser. B 65 (1995), 273-290.
14. F. Chung, S.-T. Yau, Coverings, heat kernels, and spanning trees, Electron. J. Combinatorics 6 (1999) #R12, 21 pp.
15. M. Dyer, L.A. Goldberg, M. Jerrum, The complexity of weighted Boolean #CSP,arXiv:0704.3683v1 [cs.CC] (Apr, 2007).
16. F. V. Fomin, S. Gaspers, S. Saurabh, Improved exact algorithms for counting 3- and 4-colorings, Computing and Combinatorics, 13th Annual International Conference (COCOON), Banff, Canada, July 16-19, 2007, Lecture Notes in Computer Science 4598, Springer, 2007, pp. 65-74.
17. C. M. Fortuin, P. W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models, Physica 57 (1972), 536-564.
18. H. Gebauer, Y. Okamoto, Fast exponential-time algorithms for the forest counting and the Tutte polynomial computation in graph classes, Int. J. Found. Comput. S. (2008), to appear.
19. O. Giménez, P. Hliněný, M. Noy, Computing the Tutte polynomial on graphs of bounded clique-width, SIAM J. Discrete Math. 20 (2006), 932-946.
20. C. Godsil, G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer, 2001.
21. L. A. Goldberg, M. Jerrum, Inapproximability of the Tutte polynomial, Inform. Comput. 206 (2008), 908-929.
22. G. Haggard, D. Pearce, G. Royle, Computing Tutte polynomials, Technical Report, Victoria University of Wellington, NZ, in preparation.
23. P. Hliněný, The Tutte polynomial for matroids of bounded branch-width, Combin. Probab. Comput. 15 (2006), 397-409.
24. H. Imai, Computing the invariant polynomials of graphs, networks, and matroids, IEICE T. Inf. Syst. E93-D (2000), 330-343.
25. R. Impagliazzo, R. Paturi, F. Zane, Which problems have strongly exponential complexity?, J. Comput. Syst. Sci. 63 (2001), 512-530.
26. F. Jaeger, D. L. Vertigan, D. J. A. Welsh, On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Cambridge Philos. Soc. 108 (1990), 35-53.
27. S. Kapoor, H. Ramesh, Algorithms for enumerating all spanning trees of undirected and weighted graphs, SIAM J. Comput. 24 (1995), 247-265.
28. R. M. Karp, Dynamic programming meets the principle of inclusion and exclusion, Oper. Res. Lett. 1 (1982), 49-51.
29. P. W. Kasteleyn, The statistics of dimers on a lattice: 1. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225.
30. S. Kohn, A. Gottlieb, M. Kohn, A generating function approach to the traveling salesman problem, Proceedings of the 1977 Annual Conference (ACM'77), Association for Computing Machinery, 1977, pp. 294-300.
31. M. Koivisto, Optimal 2-constraint satisfaction via sumproduct algorithms, Inform. Process. Lett. 98 (2006), 22-24.
32. W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999), 297-300.
33. E. L. Lawler, A note on the complexity of the chromatic number problem, Inf. Process. Lett. 5 (1976), 66-67.
34. S. D. Noble, Evaluating the Tutte polynomial for graphs of bounded tree-width, Combin. Probab. Comput. 7 (1998), 307-321.
35. J. G. Oxley, D. J. A. Welsh, The Tutte polynomial and percolation, Graph Theory and Related Topics (J. A. Bondy and U. S. R. Murty, Eds.), Academic Press, .1979, pp. 329-339.
36. R. B. Potts, Some generalized order-disorder transformations, Proceedings of the Cambridge Philosophical Society 48 (1952), 106-109.
37. K. Sekine, H. Imai, S. Tani, Computing the Tutte polynomial of a graph of moderate size, Algorithms and Computation, 6th International Symposium (ISAAC '95), Cairns, Australia, December 4-6, 1995, Lecture Notes in Computer Science 1004, Springer, 1995, pp. 224-233.
38. A. D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys in Combinatorics, 2005, London Mathematical Society Lecture Note Series 327, Cambridge University Press, 2005, pp. 173-226.
39. P. Traxler, The time complexity of constraint satisfaction, Proceedings of the 3rd International Workshop on Exact and Parameterized Computation (Victoria, BC, Canada, May 14-16, 2008), to appear.
40. W. T. Tutte, Graph-polynomials, Adv. Appl. Math. 32 (2004), 5-9.
41. L. G. Valiant, The complexity of enumeration and reliability problems, SIAM J. Comput. 8 (1979), 410-421.
42. D. J. A. Welsh, Complexity: Knots, Colourings and Counting, London Mathematical Society Lecture Note Series 186, Cambridge University Press, 1993.
43. D. J. A. Welsh, The Tutte polynomial, Random Structures Algorithms 15 (1999), 210-228.
44. D. J. A. Welsh, C. Merino, The Potts model and the Tutte polynomial, J. Math. Phys. 41 (2000), 1127-1152.
45. D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, 2001.
46. H. S. WiIf, Algorithms and Complexity, Prentice-Hall, 1986.
47. R. Williams, A new algorithm for optimal constraint satisfaction and its implications, Theoret. Comput. Sci. 348 (2005), 357-365.