Balancing pairs and the cross product conjecture

  title={Balancing pairs and the cross product conjecture},
  author={Graham R. Brightwell and Stefan Felsner and William T. Trotter},
In a finite partially ordered set, Prob (x>y) denotes the proportion of linear extensions in which elementx appears above elementy. In 1969, S. S. Kislitsyn conjectured that in every finite poset which is not a chain, there exists a pair (x,y) for which 1/3⩽Prob(x>y)⩽2/3. In 1984, J. Kahn and M. Saks showed that there exists a pair (x,y) with 3/11y)<8/11, but the full 1/3–2/3 conjecture remains open and has been listed among ORDER's featured unsolved problems for more than 10 years.In this… 

On the 1/3–2/3 Conjecture

The1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair, and is made progress on this conjecture by showing that it holds for certain families of posets.

Balanced pairs in partial orders

The cross-product conjecture for width two posets

. The cross–product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset P = ( X, ≺ ) with given value

Sorting under partial information (without the ellipsoid algorithm)

This work revisits the well-known problem of sorting under partial information, and develops efficient algorithms that approximate the entropy, or make sure it is computed only once, in a restricted class of graphs, permitting the use of a simpler algorithm.

Exact exponential algorithms for two poset problems

  • L. Kozma
  • Mathematics, Computer Science
  • 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

Counting Linear Extensions of a Partial Order

A partially ordered set (P,<) is a set P together with an irreflexive, transitive relation. A linear extension of (P,<) is a relation (P,≺) such that (1) for all a, b ∈ P either a ≺ b or a = b or b ≺

Sorting and Selection in Posets | SIAM Journal on Computing | Vol. 40, No. 3 | Society for Industrial and Applied Mathematics

This paper presents the first algorithm that sorts a width-w poset of size n with query complexity O(n(w + logn)) and proves that this query complexity is asymptotically optimal, and considers two related problems: finding the minimal elements, and its generalization to finding the bottom k “levels,” called the k-selection problem.

Efficient computation of rank probabilities in posets

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.

Improving the $$\frac{1}{3} - \frac{2}{3}$$ Conjecture for Width Two Posets

A sequence of posets T n of width 2 is constructed with δ ( T n ) → β ≈ 0.348843…, giving an improvement over a construction of Chen (2017) and over the finite posets found by Peczarski (2017).



Balancing Pairs in Partially Ordered Sets

J. Kahn and M. Saks proved that if P is a partially ordered set and is not a chain, then there exists a pair x; y 2 P so that the number of linear extensions of P with x less than y is at least 3=11

Semiorders and the 1/3–2/3 conjecture

A well-known conjecture of Fredman is that, for every finite partially ordered set (X, <) which is not a chain, there is a pair of elements x, y such that P(x<y), the proportion of linear extensions

Linear extensions of infinite posets

A correlational inequality for linear extensions of a poset

AbstractSuppose 1, 2, and 3 are pairwise incomparable points in a poset onn≥3 points. LetN (ijk) be the number of linear extensions of the poset in whichi precedesj andj precedesk. Define λ by

Balancing poset extensions

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.

How Good is the Information Theory Bound in Sorting?

  • M. Fredman
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1976

The 1/3-2/3 Conjecture for 5-Thin Posets

The 1/3–2/3 conjecture is established in the case where every element of the poser is incomparable with at most five others, and the proof involves the use of a computer to eliminate a large number of cases.

The FKG Inequality and Some Monotonicity Properties of Partial Orders

  • L. Shepp
  • Mathematics
    SIAM J. Algebraic Discret. Methods
  • 1980
A simple example is given to show that the more general inequality (*) where P is allowed to contain inequalities of the form $a_i < b_j $ is false, which is surprising because as Graham, Yao, and Yao proved, the general inequality(*) does hold if P totally orders both the a's and the b’s separately.

An inequality for the weights of two families of sets, their unions and intersections

then ~(A) fi(B) < 7(A v B) cS(A A B) for all A, B ~ S, (2) where e(A) = ~(a~A) e(a) and A v B = {awb; aeA, b~B} and A A B = {ac~b; a~A, b~B}. Since every distributive lattice can be embedded in the

The Information-Theoretic Bound is Good for Merging

  • N. Linial
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1984
There exists an algorithm which will take no more than C\log _2 N comparisons where C = (\log_2 ((\sqrt 5 + 1)/ 2))^{ -1} $ and the constant C is best possible.