Range of the Fractional Weak Discrepancy Function

In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset and characterize semiorders in terms of these values. In [6], we defined the fractional weak discrepancy wdF (P ) of a poset P = (V,≺) to be 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. This notion builds on previous work on weak discrepancy in [3, 7, 8]. We prove here that the range of values of the function wdF is the set of rational numbers that are either at least one or equal to r r+1 for some nonnegative integer r. Moreover, P is a semiorder if and only if wdF (P ) < 1, and the range taken over all semiorders is the set of such fractions r r+1 .