## Abstract

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## Preview

Unable to display preview. Download preview PDF.

## References

- 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.M. D.Atkinson (1985) Partial orders and comparison problems,
*Congressus Numerantium***47**, 77–88.Google Scholar - 3.M. D.Atkinson and H. W.Chang (1985) Extensions of partial orders of bounded width,
*Congressus Numerantium***52**, 21–35.Google Scholar - 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.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.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.M. Dyer and A. Frieze, On the complexity of computing the volume of a polyhedron,
*SIAM J. Computing*, to appear.Google Scholar - 8.M. Dyer and A. Frieze, Computing the volume of convex bodies: a case where randomness provably helps, preprint.Google Scholar
- 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.J. Feigenbaum, private communication.Google Scholar
- 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.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.G. H. Hardy and E. M. Wright (1960)
*An Introduction to the Theory of Numbers*, 4th Ed., Oxford University Press.Google Scholar - 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.
- 16.A.Karzanov and L.Khachiyan (1991) On the conductance of order Markov chains,
*Order***8**(1), 7–15.Google Scholar - 17.L. Khachiyan, Complexity of polytope volume computation,
*Recent Progress in Discrete Computational Geometry*, J. Pach ed., Springer-Verlag, to appear.Google Scholar - 18.H. Kierstead and W. T. Trotter, The number of depth-first searches of an ordered set, submitted.Google Scholar
- 19.N.Linial (1986) Hard enumeration problems in geometry and combinatorics,
*SIAM J. Alg. Disc. Meth.*7(2), 331–335.Google Scholar - 20.L.Lovász (1986)
*An Algorithmic Theory of Numbers, Graphs and Convexity*, SIAM, Philadelphia.Google Scholar - 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.P.Matthews (1991) Generating a random linear extension of a partial order,
*Annals of Probability*,**19**, 1367–1392.Google Scholar - 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.A.Sinclair and M.Jerrum (1989) Approximate counting, generation and rapidly mixing Markov chains,
*Information and Computation***82**, 93–133.Google Scholar - 25.G. Steiner, Polynomial algorithms to count linear extensions in certain posets,
*Congressus Numerantium*, to appear.Google Scholar - 26.G. Steiner, On counting constrained depth-first linear extensions of ordered sets, preprint.Google Scholar
- 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.L. G.Valiant (1979) The complexity of computing the permanent,
*Theoret. Comput. Sci.***8**, 189–201.CrossRefGoogle Scholar - 29.L. G.Valiant (1979) The complexity of enumeration and reliability problems,
*SIAM J. Comput*.**8**, 410–421.Google Scholar - 30.P.Winkler (1982) Average height in a partially orderd set,
*Discrete Math.***39**, 337–341.CrossRefGoogle Scholar

## Copyright information

© Kluwer Academic Publishers 1991