The total weak discrepancy of a partially ordered set

Abstract

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.