Counting linear extensions Graham Brightwell Peter Winkler Article Received: 06 June 1991 Accepted: 25 October 1991 DOI :
10.1007/BF00383444

Cite this article as: Brightwell, G. & Winkler, P. Order (1991) 8: 225. doi:10.1007/BF00383444
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 Research carried out while this author was visiting Bellcore under the auspices of DIMACS.

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.

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.

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.

7.

M. Dyer and A. Frieze, On the complexity of computing the volume of a polyhedron, SIAM J. Computing , to appear.

8.

M. Dyer and A. Frieze, Computing the volume of convex bodies: a case where randomness provably helps, preprint.

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.

10.

J. Feigenbaum, private communication.

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.

CrossRef Google Scholar 13.

G. H. Hardy and E. M. Wright (1960) An Introduction to the Theory of Numbers , 4th Ed., Oxford University Press.

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.

15.

J.Kahn and M.Saks (1984) Balancing poset extensions,

Order
1 (2), 113–126.

Google Scholar 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.

18.

H. Kierstead and W. T. Trotter, The number of depth-first searches of an ordered set, submitted.

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.

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.

26.

G. Steiner, On counting constrained depth-first linear extensions of ordered sets, preprint.

27.

S. Toda (1989) On the computational power of PP and +P, Proc. 30th IEEE Symposium on Foundations of Computer Science , 514–519.

28.

L. G.Valiant (1979) The complexity of computing the permanent,

Theoret. Comput. Sci.
8 , 189–201.

CrossRef Google 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.

CrossRef Google Scholar © Kluwer Academic Publishers 1991

Authors and Affiliations Graham Brightwell Peter Winkler 1. London School of Economics and Political Science London UK 2. Bellcore Morristown USA