, Volume 8, Issue 3, pp 225–242 | Cite as

Counting linear extensions

  • Graham Brightwell
  • Peter Winkler


We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is contrasted with recent work giving randomized polynomial-time algorithms for estimating the number of linear extensions.

One consequence of our main result is that computing the volume of a rational polyhedron is strongly #P-hard. We also show that the closely related problems of determining the average height of an element x of a give poset, and of determining the probability that x lies below y in a random linear extension, are #P-complete.

AMS subject classifications (1991)

06A06 68C25 

Key words

Partial order linear extension #P-complete 


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  1. 1.
    D. Applegate and R. Kannan (1991) Sampling and integration of near log-concave functions, Proc. 23rd ACM Symposium on the Theory of Computing, 156–163.Google Scholar
  2. 2.
    M. D.Atkinson (1985) Partial orders and comparison problems, Congressus Numerantium 47, 77–88.Google Scholar
  3. 3.
    M. D.Atkinson and H. W.Chang (1985) Extensions of partial orders of bounded width, Congressus Numerantium 52, 21–35.Google Scholar
  4. 4.
    M. D.Atkinson and H. W.Chang (1987) Computing the number of mergings with constraints, Information Processing Letters 24, 289–292.Google Scholar
  5. 5.
    G. Brightwell and P. Winkler (1991) Counting linear extensions is #P-complete, Proc. 23rd ACM Symposium on the Theory of Computing, 175–181.Google Scholar
  6. 6.
    A. Z. Broder (1986) How hard is it to marry at random? (On the approximation of the permanent), Proc. 18th ACM Symposium on the Theory of Computing, 50–58.Google Scholar
  7. 7.
    M. Dyer and A. Frieze, On the complexity of computing the volume of a polyhedron, SIAM J. Computing, to appear.Google Scholar
  8. 8.
    M. Dyer and A. Frieze, Computing the volume of convex bodies: a case where randomness provably helps, preprint.Google Scholar
  9. 9.
    M. Dyer, A. Frieze, and R. Kannan (1989) A randomly polynomial time algorithm for estimating volumes of convex bodies, Proc 21st ACM Symposium on the Theory of Computing, 375–381.Google Scholar
  10. 10.
    J. Feigenbaum, private communication.Google Scholar
  11. 11.
    P. C.Fishburn and W. V.Gehrlein (1975) A comparative analysis of methods for constructing weak orders from partial orders. J. Math. Sociology 4, 93–102.Google Scholar
  12. 12.
    M.Habib and R. H.Mohring (1987) On some complexity properties of N-free posets and posets with bounded decomposition diameter, Discrete Math. 63, 157–182.CrossRefGoogle Scholar
  13. 13.
    G. H. Hardy and E. M. Wright (1960) An Introduction to the Theory of Numbers, 4th Ed., Oxford University Press.Google Scholar
  14. 14.
    M. Jerrum and A. Sinclair (1988) Conductance and the rapid mixing property for Markov chains: the approximation of the permanent resolved, Proceedings of the 20th ACM Symposium on Theory of Computing, 235–244.Google Scholar
  15. 15.
    J.Kahn and M.Saks (1984) Balancing poset extensions, Order 1(2), 113–126.Google Scholar
  16. 16.
    A.Karzanov and L.Khachiyan (1991) On the conductance of order Markov chains, Order 8 (1), 7–15.Google Scholar
  17. 17.
    L. Khachiyan, Complexity of polytope volume computation, Recent Progress in Discrete Computational Geometry, J. Pach ed., Springer-Verlag, to appear.Google Scholar
  18. 18.
    H. Kierstead and W. T. Trotter, The number of depth-first searches of an ordered set, submitted.Google Scholar
  19. 19.
    N.Linial (1986) Hard enumeration problems in geometry and combinatorics, SIAM J. Alg. Disc. Meth. 7(2), 331–335.Google Scholar
  20. 20.
    L.Lovász (1986) An Algorithmic Theory of Numbers, Graphs and Convexity, SIAM, Philadelphia.Google Scholar
  21. 21.
    L. Lovász and M. Simonovits (1990) The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume, Proc. 31st IEEE Symposium on Foundations of Computer Science, 346–355.Google Scholar
  22. 22.
    P.Matthews (1991) Generating a random linear extension of a partial order, Annals of Probability, 19, 1367–1392.Google Scholar
  23. 23.
    S.Provan and M. O.Ball (1983) On the complexity of counting cuts and of computing the probability that a graph is connected, SIAM J. Computing 12, 777–788.Google Scholar
  24. 24.
    A.Sinclair and M.Jerrum (1989) Approximate counting, generation and rapidly mixing Markov chains, Information and Computation 82, 93–133.Google Scholar
  25. 25.
    G. Steiner, Polynomial algorithms to count linear extensions in certain posets, Congressus Numerantium, to appear.Google Scholar
  26. 26.
    G. Steiner, On counting constrained depth-first linear extensions of ordered sets, preprint.Google Scholar
  27. 27.
    S. Toda (1989) On the computational power of PP and +P, Proc. 30th IEEE Symposium on Foundations of Computer Science, 514–519.Google Scholar
  28. 28.
    L. G.Valiant (1979) The complexity of computing the permanent, Theoret. Comput. Sci. 8, 189–201.CrossRefGoogle Scholar
  29. 29.
    L. G.Valiant (1979) The complexity of enumeration and reliability problems, SIAM J. Comput. 8, 410–421.Google Scholar
  30. 30.
    P.Winkler (1982) Average height in a partially orderd set, Discrete Math. 39, 337–341.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Graham Brightwell
    • 1
  • Peter Winkler
    • 2
  1. 1.London School of Economics and Political ScienceLondonUK
  2. 2.BellcoreMorristownUSA

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