Computer Science, Mathematics
The fractional weak discrepancy wdF(P) of a poset P=(V,≺) was introduced in Shuchat et al. (2007)  as the minimum nonnegative k for which there exists a function f:V→R satisfying (i) if a≺b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009)  and  on the range of wdF for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0≤s−1≤r<2s.
Shuchat, A., Shull, R., & Trenk, A. N. (2011). Fractional Weak Discrepancy and Split Semiorders. Discrete Applied Mathematics, 159(7): 647-660. doi:10.1016/j.dam.2010.04.014