The fractional weak discrepancy of a partially ordered set

In this paper we introduce the notion of the fractional weak discrepancy of a poset, building on previous work on weak discrepancy in [5, 8, 9]. The fractional weak discrepancy wdF (P ) of a poset P = (V,≺) is the minimum nonnegative k for which there exists a function f : V → R satisfying (1) if a ≺ b then f(a) + 1 ≤ f(b) and (2) if a ‖ b then |f(a)− f(b)| ≤ k. We formulate the fractional weak discrepancy problem as a linear program and show how its solution can also be used to calculate the (integral) weak discrepancy. We interpret the dual linear program as a circulation problem in a related directed graph and use this to give a structural characterization of the fractional weak discrepancy of a poset.