Computer Science, Mathematics
The fractional weak discrepancy wdF(P) of a poset P=(V,≺) was introduced in [A. Shuchat, R. Shull, A. Trenk, The fractional weak discrepancy of a partially ordered set, Discrete Applied Mathematics 155 (2007) 2227–2235] as the minimum nonnegative k for which there exists a function 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 [A. Shuchat, R. Shull, A. Trenk, Range of the fractional weak discrepancy function, ORDER 23 (2006) 51–63; A. Shuchat, R. Shull, A. Trenk, Fractional weak discrepancy of posets and certain forbidden configurations, in: S.J. Brams, W.V. Gehrlein, F.S. Roberts (Eds.), The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishburn, Springer, New York, 2009, pp. 291–302] on the range of the wdF function for semiorders (interval orders with no induced ) to interval orders with no , where n≥3. In particular, we prove that the range for such posets P is the set of rationals that can be written as r/s, where 0≤s−1≤r<(n−2)s. If wdF(P)=r/s and P has an optimal forcing cycle C with and , then r≤(n−2)(s−1). Moreover when s≥2, for each r satisfying s−1≤r≤(n−2)(s−1) there is an interval order having such an optimal forcing cycle and containing no.
Shuchat, A., Shull, R., & Trenk, A. N. (2009). Fractional Weak Discrepancy and Interval Orders. Discrete Applied Mathematics, 157(8): 1873–1884. doi:10.1016/j.dam.2008.12.011