Average relational distance in linear extensions of posets

Brightwell, Graham; Patel, Viresh

doi:10.1016/j.disc.2009.10.016

Cited by 2 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The linear discrepancy of a poset, defined formally in [12], is equivalent to the weak discrepancy with the additional condition in Definition 1.2 that the labeling function be injective. Similarly, total linear discrepancy, studied in [2] and [5], is equivalent to total weak discrepancy with an injective labeling function.…”

Section: Comparability Invariantsmentioning

confidence: 99%

“…Likewise, total linear discrepancy is not a comparability invariant. Using the results of [2] and [5], it is easy to check that the total linear discrepancy of P is 8, while that of Q is 7.…”

Section: Comparability Invariantsmentioning

confidence: 99%

“…The incomparabilities can be written as a i a j , i < j, corresponding to the (i, j) pairs (1, 4), (1,5), (1,6), (2,6), (3,6), (4,5), (4,6), (5,6).…”

Section: Fractional Total Weak Discrepancy and Linear Programmingmentioning

confidence: 99%

“…We do not show the steps for the poset Z of Figure 1, but some of the ranges reduce to intervals with positive length. In particular, I(a 3 ) = [−1, 0], I(a 4 ) = [0, 1], and I(a 5 ) = [1,2]. Choosing f (a) = r(a) in all cases produces the integer labeling on the right side of Figure 1.…”

Section: Fractional Total Weak Discrepancy Equals Total Weak Discrepancymentioning

confidence: 99%

“…The labeling functions that are optimal for the total linear discrepancy of a poset are characterized in [2] and [5]. We can pose a similar problem for total weak discrepancy.…”

Section: Open Questionsmentioning

confidence: 99%

See 4 more Smart Citations

The total weak discrepancy of a partially ordered set

2011

View full text Add to dashboard Cite

We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : V → Z satisfying (i) if a ≺ b then f (a) + 1 ≤ f (b) and (ii) |f (a) − f (b)| ≤ k, where the sum is taken over all unordered pairs {a, b} of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P. These concepts are related to the notions of weak and fractional weak discrepancy, where (ii) must hold not for the sum but for each individual pair of incomparable elements of P. We prove that, unlike the latter, the total weak and fractional total weak discrepancy of P are always the same, and we give a polynomial-time algorithm to find their common value. We use linear programming duality and complementary slackness to obtain this result.

show abstract

Section: Comparability Invariantsmentioning

confidence: 99%

Section: Comparability Invariantsmentioning

confidence: 99%

“…The incomparabilities can be written as a i a j , i < j, corresponding to the (i, j) pairs (1, 4), (1,5), (1,6), (2,6), (3,6), (4,5), (4,6), (5,6).…”

Section: Fractional Total Weak Discrepancy and Linear Programmingmentioning

confidence: 99%

Section: Fractional Total Weak Discrepancy Equals Total Weak Discrepancymentioning

confidence: 99%

“…The labeling functions that are optimal for the total linear discrepancy of a poset are characterized in [2] and [5]. We can pose a similar problem for total weak discrepancy.…”

Section: Open Questionsmentioning

confidence: 99%

See 3 more Smart Citations