Counting linear extensions
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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.
- 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.
- Atkinson, M. D. (1985) Partial orders and comparison problems. Congressus Numerantium 47: pp. 77-88
- Atkinson, M. D., Chang, H. W. (1985) Extensions of partial orders of bounded width. Congressus Numerantium 52: pp. 21-35
- Atkinson, M. D., Chang, H. W. (1987) Computing the number of mergings with constraints. Information Processing Letters 24: pp. 289-292
- G. Brightwell and P. Winkler (1991) Counting linear extensions is #P-complete, 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. 93-102
- Habib, M., Mohring, R. H. (1987) On some complexity properties of N-free posets and posets with bounded decomposition diameter. Discrete Math. 63: pp. 157-182 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. 113-126
- Karzanov, A., Khachiyan, L. (1991) On the conductance of order Markov chains. Order 8: pp. 7-15
- L. Khachiyan, Complexity of polytope volume computation, Recent Progress in Discrete Computational Geometry, J. Pach ed., Springer-Verlag, to appear.
- H. Kierstead and W. T. Trotter, The number of depth-first searches of an ordered set, submitted.
- Linial, N. (1986) Hard enumeration problems in geometry and combinatorics. SIAM J. Alg. Disc. Meth. 7: pp. 331-335
- 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. 1367-1392
- 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. 777-788
- Sinclair, A., Jerrum, M. (1989) Approximate counting, generation and rapidly mixing Markov chains. Information and Computation 82: pp. 93-133
- G. Steiner, Polynomial algorithms to count linear extensions in certain posets, Congressus Numerantium, to appear.
- G. Steiner, On counting constrained depth-first 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. 189-201 CrossRef
- Valiant, L. G. (1979) The complexity of enumeration and reliability problems. SIAM J. Comput 8: pp. 410-421
- Winkler, P. (1982) Average height in a partially orderd set. Discrete Math. 39: pp. 337-341 CrossRef
- Counting linear extensions
Volume 8, Issue 3 , pp 225-242
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