Computer Science, Mathematics
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 , 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 [over] 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 [over] r+1.
Shuchat, A., Shull, R., & Trenk, A. N. (2006). Range of the Fractional Weak Discrepancy Function. Order 23: 51–63. DOI 10.1007/s11083-006-9030-4