The total weak discrepancy of a partially ordered set

Alan Shuchat, Randy Shull, Ann N. Trenk

Abstract


We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : VZ satisfying (i) if a \prec b then f(a) + 1 ≤ f(b) and (ii) Σ|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.

Keywords


Partial orders; weak discrepancy; graph algorithms; network models; linear programming duality

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DOI: https://doi.org/10.26493/1855-3974.159.1f8

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications