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 [J.G. Gimbel and A.N. Trenk, On the weakness of an ordered set, SIAM J. Discrete Math. 11 (1998) 655–663; P.J. Tanenbaum, A.N. Trenk, P.C. Fishburn, Linear discrepancy and weak discrepancy of partially ordered sets, ORDER 18 (2001) 201–225; A.N. Trenk, On k-weak orders: recognition and a tolerance result, Discrete Math. 181 (1998) 223–237]. The fractional weak discrepancy  wdF(P)wdF(P) of a poset P=(V,≺)P=(V,≺) is the minimum nonnegative k   for which there exists a function f:V→Rf:V→R satisfying (1) if a≺ba≺b then f(a)+1⩽f(b)f(a)+1⩽f(b) and (2) if a∥ba∥b then |f(a)-f(b)|⩽k|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.