The total linear discrepancy of an ordered set

Howard, David M.; Shull, Randy; Streib, Noah; Trenk, Ann N.

doi:10.1016/j.disc.2009.10.015

Cited by 2 publications

(4 citation statements)

References 9 publications

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“…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%

“…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%

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The total weak discrepancy of a partially ordered set

2011

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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.

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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%

“…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%

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