Negative results on counting

  • L. G. Valiant
Part of the Lecture Notes in Computer Science book series (LNCS, volume 67)


Span Tree Bipartite Graph Perfect Matchings Hamiltonian Path Search Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    A.V. Aho, J.E. Hopcroft and J.D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass. (1974).Google Scholar
  2. [2]
    D. Angluin. On counting problems and the polynomial time hierarchy, to appear (1978).Google Scholar
  3. [3]
    T. Baker, J. Gill and R. Solovay. Relativizations of the P = ? NP question. SIAM J. Comput. 4(1975) 431–442.CrossRefGoogle Scholar
  4. [4]
    M.N. Barber and B.W. Ninham. Random and restricted walks. Gordon and Breach, New York (1970).Google Scholar
  5. [5]
    C. Berge. Graphs and Hypergraphs. North-Holland, Amsterdam (1973).Google Scholar
  6. [6]
    A. Borodin and I. Munro. The Computational Complexity of Algebraic and Numeric Problems. American Elsevier, New York (1975).Google Scholar
  7. [7]
    N.G. De Bruijn. Polya's theory of counting. In Applied Combinatorial Mathematics, E.F. Beckenbach (ed), Wiley, New York (1964).Google Scholar
  8. [8]
    A. Cayley. On the theory of the analytical form called trees. Collected Works, Vol.3, Cambridge Univ. Press (1890) 243–246.Google Scholar
  9. [9]
    S.A. Cook. The complexity of theorem proving procedures. Proc. 3rd ACM Symp. on Theory of Computing (1971) 151–158.Google Scholar
  10. [10]
    H.L. Frisch and J.M. Hammersley. Percolation processes and related topics. J. SIAM Appl. Math. 11 (1963) 894–918.CrossRefGoogle Scholar
  11. [11]
    H.N. Gabow. Finding all spanning trees of undirected and directed graphs. Tech. Report, Univ. of Colorado, Boulder (1977).Google Scholar
  12. [12]
    H.S. Green and G.A. Hurst. Order-Disorder Phenomena. Interscience. London (1964).Google Scholar
  13. [13]
    J.M. Hammersley and D.J.A. Welsh. Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford Ser. 2, 13 (1962) 108–110.Google Scholar
  14. [14]
    F. Harary and E.M. Palmer. Graphical Enumeration. Academic Press. (1973).Google Scholar
  15. [15]
    A. Itai, M. Rodeh and S.L. Tanimoto. Some matching problems for bipartite graphs. JACM 25 (1978) 517–525.CrossRefGoogle Scholar
  16. [16]
    R.M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, R.E. Miller and J.W. Thatcher (eds). Plenum Press, New York (1972).Google Scholar
  17. [17]
    P.W. Kasteleyn. Dimer statistics and phase transitions. J.Math. Phys. 4 (1963) 287–293.CrossRefGoogle Scholar
  18. [18]
    P.W. Kasteleyn. A soluble self-avoiding walk problem. Physica 29 (1963) 1329–1337.Google Scholar
  19. [19]
    P.W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics. F. Harary (ed.) Academic Press (1967).Google Scholar
  20. [20]
    G. Kirchhoff. Über die Ausflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Verleilung galvanischer Ströme gefuhrt wird. Ann. Phys. Chem. 72 (1847) 497–508.Google Scholar
  21. [21]
    D.E. Knuth. Estimating the efficiency of backtrack algorithms. Math. Comp. 29 (1975) 121–136.Google Scholar
  22. [22]
    C.L. Liu. Introduction to Combinatorial Mathematics. McGraw Hill, New York (1968).Google Scholar
  23. [23]
    R. Mathon. A note on the graph isomorphism counting problem, to appear (1977).Google Scholar
  24. [24]
    J.K. Percus. Combinatorial Methods. Springer-Verlag, New York (1971).Google Scholar
  25. [25]
    G. Pólya. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acto Math 68 (1937) 145–154.Google Scholar
  26. [26]
    R.E. Tarjan. Enumeration of the elementary circuits of a directed graph. SIAM J. on Comput 2 (1973) 211–216.CrossRefGoogle Scholar
  27. [27]
    W.T. Tutte. The dissection of equilateral triangles into equilateral triangles. Proc. Cambridge Phil. Soc. 44 (1948) 463–482.Google Scholar
  28. [28]
    L.G. Valiant. The complexity of computing the permanent. Tech. Rep. CSR-14-77, Edinburgh Univ. (1977). Also, TCS, to appear.Google Scholar
  29. [29]
    L.G. Valiant. The complexity of enumeration and reliability problems. Tech. Rep. CSR-15-77, Edinburgh Univ. (1977). Also, SIAM J. on Comput. to appear.Google Scholar
  30. [30]
    L.G. Valiant. Completeness classes in algebra (1978), to appear.Google Scholar
  31. [31]
    T. van der Aardenne-Ehrenfest and N.G. de Bruijn. Circuits and trees in oriented line graphs. Simon Stevin 28 (1951) 203–217.Google Scholar
  32. [32]
    D.J.A. Welsh. Percolation and related topics. Sci. Prog. Oxf. (1977) 64, 65–83.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1979

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  • L. G. Valiant

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