#### Department

Computer Science, Mathematics

#### Document Type

Article

#### Publication Date

10-2007

#### Abstract

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 **wd*_{F}(P) of a poset P=(V,≺) is 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. 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.

#### Recommended Citation

Shuchat, A., Shull, R., & Trenk, A. N. (2007). The fractional weak discrepancy of a partially ordered set. Discrete Applied Mathematics, 155 (17): 2227–2235. doi:10.1016/j.dam.2007.05.032

#### Version

Post-print

## Comments

Post-print version, April 17, 2007. Published version in: Discrete Applied Mathematics, 155 (17): 2227–2235 (2007). doi:10.1016/j.dam.2007.05.032