# Counting linear extensions

@article{Brightwell1991CountingLE, title={Counting linear extensions}, author={Graham R. Brightwell and Peter Winkler}, journal={Order}, year={1991}, volume={8}, pages={225-242} }

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…

## 276 Citations

### CS 294 Partition Functions Fall 2020

- Computer Science, Mathematics
- 2020

It turns out that almost all interesting problems of this type are hard, and this even applies to computing partition functions Z(λ) at any specific value of λ (except for one or two trivial values).

### Counting Linear Extensions of Sparse Posets

- Computer ScienceIJCAI
- 2016

Two algorithms for computing the number of linear extensions of a given n-element poset based on variable elimination via inclusion-exclusion and runs in time O(nt+4), where t is the treewidth of the cover graph.

### Exploiting the Lattice of Ideals Representation of a Poset

- Mathematics, Computer ScienceFundam. Informaticae
- 2006

In this paper, we demonstrate how some simple graph counting operations on the ideal lattice representation of a partially ordered set (poset)P allow for the counting of the number of linear…

### Exact exponential algorithms for two poset problems

- Mathematics, Computer ScienceSWAT
- 2020

Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-$n$ poset, is to compute…

### Faster random generation of linear extensions

- Mathematics, Computer ScienceSODA '98
- 1998

### Efficient computation of rank probabilities in posets

- Mathematics, Computer Science
- 2009

An algorithm that can be used to sample weak order extensions uniformly at random is introduced and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity.

### A loop-free algorithm for generating the linear extensions of a poset

- Mathematics
- 1995

A precise concept of when a combinatorial counting problem is “hard” was first introduced by Valiant (1979) when he defined the notion of a #P-complete problem. Correspondingly, there has been…

### Counting Linear Extensions of Restricted Posets

- MathematicsElectron. J. Comb.
- 2020

It is proved that the number of linear extension for posets of height two is #P-complete and that this holds for incidence poset of graphs and for incidencePosets of graphs with certain restrictions.

### Approximating the Volume of Tropical Polytopes is Difficult

- MathematicsInt. J. Algebra Comput.
- 2019

It is deduced that there is no approximation algorithm of factor for counting the number of integer points in tropical polytopes described by vertices, and it follows that approximating these values for tropicalpolytopes is more difficult than for classical poly topes.

## References

SHOWING 1-10 OF 24 REFERENCES

### Balancing poset extensions

- Mathematics, Computer Science
- 1984

It is shown that any finite partially ordered setP contains a pair of elementsx andy such that the proportion of linear extensions ofP in whichx lies belowy is between 3/11 and 8/11, so that the information theoretic lower bound for sorting under partial information is tight up to a multiplicative constant.

### The Complexity of Enumeration and Reliability Problems

- Mathematics, Computer ScienceSIAM J. Comput.
- 1979

For a large number of natural counting problems for which there was no previous indication of intractability, that they belong to the class of computationally eqivalent counting problems that are at least as difficult as the NP-complete problems.

### On some complexity properties of N-free posets and posets with bounded decomposition diameter

- MathematicsDiscret. Math.
- 1987

### Algorithmic theory of numbers, graphs and convexity

- MathematicsCBMS-NSF regional conference series in applied mathematics
- 1986

How to Round Numbers Preliminaries and some Applications in Combinatorics Cuts and Joins Chromatic Number, Cliques and Perfect Graphs Minimizing a Submodular Function.

### On the conductance of order Markov chains

- Mathematics
- 1991

Let Q be a convex solid in ℝn, partitioned into two volumes u and v by an area s. We show that s>min(u,v)/diam Q, and use this inequality to obtain the lower bound n-5/2 on the conductance of order…

### Hard enumeration problems in geometry and combinatorics

- Mathematics
- 1986

A number of natural enumeration problems in geometry and combinatorics are shown to be complete in the class # P introduced by Valiant. Among others this is established for the numeration of vertices…

### The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected

- Mathematics, Computer ScienceSIAM J. Comput.
- 1983

Several enumeration and reliability problems are shown to be # P-complete, and hence, at least as hard as NP-complete problems. Included are important problems in network reliability analysis,…

### The Complexity of Computing the Permanent

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1979

### On the computational power of PP and (+)P

- Computer Science30th Annual Symposium on Foundations of Computer Science
- 1989

It follows that neither PP nor (+)P is a subset of or equivalent to PH unless PH collapses to a finite level, strong evidence that both classes are strictly harder than PH.