Counting linear extensions

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.

[1]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[2]  Peter C. Fishburn,et al.  A comparative analysis of methods for constructing weak orders from partial orders , 1975 .

[3]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[4]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[5]  Peter Winkler Average height in a partially ordered set , 1982, Discret. Math..

[6]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[7]  J. Kahn,et al.  Balancing poset extensions , 1984 .

[8]  Andrei Z. Broder,et al.  How hard is it to marry at random? (On the approximation of the permanent) , 1986, STOC '86.

[9]  N. Linial Hard enumeration problems in geometry and combinatorics , 1986 .

[10]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[11]  Mike D. Atkinson,et al.  Computing the Number of Mergings with Constraints , 1987, Inf. Process. Lett..

[12]  Rolf H. Möhring,et al.  On some complexity properties of N-free posets and posets with bounded decomposition diameter , 1987, Discret. Math..

[13]  Mark Jerrum,et al.  Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved , 1988, STOC '88.

[14]  Martin E. Dyer,et al.  On the Complexity of Computing the Volume of a Polyhedron , 1988, SIAM J. Comput..

[15]  Martin E. Dyer,et al.  A Random Polynomial Time Algorithm for Approximating the Volume of Convex Bodies , 1989, STOC.

[16]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[17]  W. Trotter,et al.  The number of depth-first searches of an ordered set , 1989 .

[18]  Seinosuke Toda On the computational power of PP and (+)P , 1989, 30th Annual Symposium on Foundations of Computer Science.

[19]  Miklós Simonovits,et al.  The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[20]  P. Matthews Generating a Random Linear Extension of a Partial Order , 1991 .

[21]  Peter Winkler,et al.  Counting linear extensions is #P-complete , 1991, STOC '91.

[22]  M. Dyer Computing the volume of convex bodies : a case where randomness provably helps , 1991 .

[23]  L. Khachiyan,et al.  On the conductance of order Markov chains , 1991 .

[24]  David Applegate,et al.  Sampling and integration of near log-concave functions , 1991, STOC '91.