Find out how to access previewonly content
Counting linear extensions
 Graham Brightwell,
 Peter Winkler
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #Pcomplete, settling a longstanding open question. This result is contrasted with recent work giving randomized polynomialtime 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 #Phard. 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 #Pcomplete.
Research carried out while this author was visiting Bellcore under the auspices of DIMACS.
Communicated by I. Rival
 D. Applegate and R. Kannan (1991) Sampling and integration of near logconcave functions, Proc. 23rd ACM Symposium on the Theory of Computing, 156–163.
 Atkinson, M. D. (1985) Partial orders and comparison problems. Congressus Numerantium 47: pp. 7788
 Atkinson, M. D., Chang, H. W. (1985) Extensions of partial orders of bounded width. Congressus Numerantium 52: pp. 2135
 Atkinson, M. D., Chang, H. W. (1987) Computing the number of mergings with constraints. Information Processing Letters 24: pp. 289292
 G. Brightwell and P. Winkler (1991) Counting linear extensions is #Pcomplete, Proc. 23rd ACM Symposium on the Theory of Computing, 175–181.
 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.
 M. Dyer and A. Frieze, On the complexity of computing the volume of a polyhedron, SIAM J. Computing, to appear.
 M. Dyer and A. Frieze, Computing the volume of convex bodies: a case where randomness provably helps, preprint.
 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.
 J. Feigenbaum, private communication.
 Fishburn, P. C., Gehrlein, W. V. (1975) A comparative analysis of methods for constructing weak orders from partial orders. J. Math. Sociology 4: pp. 93102
 Habib, M., Mohring, R. H. (1987) On some complexity properties of Nfree posets and posets with bounded decomposition diameter. Discrete Math. 63: pp. 157182 CrossRef
 G. H. Hardy and E. M. Wright (1960) An Introduction to the Theory of Numbers, 4th Ed., Oxford University Press.
 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.
 Kahn, J., Saks, M. (1984) Balancing poset extensions. Order 1: pp. 113126
 Karzanov, A., Khachiyan, L. (1991) On the conductance of order Markov chains. Order 8: pp. 715
 L. Khachiyan, Complexity of polytope volume computation, Recent Progress in Discrete Computational Geometry, J. Pach ed., SpringerVerlag, to appear.
 H. Kierstead and W. T. Trotter, The number of depthfirst searches of an ordered set, submitted.
 Linial, N. (1986) Hard enumeration problems in geometry and combinatorics. SIAM J. Alg. Disc. Meth. 7: pp. 331335
 Lovász, L. (1986) An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM, Philadelphia
 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.
 Matthews, P. (1991) Generating a random linear extension of a partial order. Annals of Probability 19: pp. 13671392
 Provan, S., Ball, M. O. (1983) On the complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Computing 12: pp. 777788
 Sinclair, A., Jerrum, M. (1989) Approximate counting, generation and rapidly mixing Markov chains. Information and Computation 82: pp. 93133
 G. Steiner, Polynomial algorithms to count linear extensions in certain posets, Congressus Numerantium, to appear.
 G. Steiner, On counting constrained depthfirst linear extensions of ordered sets, preprint.
 S. Toda (1989) On the computational power of PP and +P, Proc. 30th IEEE Symposium on Foundations of Computer Science, 514–519.
 Valiant, L. G. (1979) The complexity of computing the permanent. Theoret. Comput. Sci. 8: pp. 189201 CrossRef
 Valiant, L. G. (1979) The complexity of enumeration and reliability problems. SIAM J. Comput 8: pp. 410421
 Winkler, P. (1982) Average height in a partially orderd set. Discrete Math. 39: pp. 337341 CrossRef
 Title
 Counting linear extensions
 Journal

Order
Volume 8, Issue 3 , pp 225242
 Cover Date
 19910901
 DOI
 10.1007/BF00383444
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 06A06
 68C25
 Partial order
 linear extension
 #Pcomplete
 Authors

 Graham Brightwell ^{(1)}
 Peter Winkler ^{(2)}
 Author Affiliations

 1. London School of Economics and Political Science, Houghton Street, London, UK
 2. Bellcore, 445 South St., Morristown, New Jersey, USA