#### Department

Computer Science, Mathematics

#### Document Type

Article

#### Publication Date

3-15-2011

#### Abstract

We define the total weak discrepancy of a poset *P* as the minimum nonnegative integer *k* for which there exists a function *f* : *V* → **Z** satisfying (i) if *a* \prec *b* then *f*(*a*) + 1 ≤ *f*(*b*) and (ii) Σ|*f*(*a*) − *f*(*b*)| ≤ *k*, where the sum is taken over all unordered pairs {*a*, *b*} of incomparable elements. If we allow *k* and *f* to take real values, we call the minimum *k* the fractional total weak discrepancy of *P*. These concepts are related to the notions of weak and fractional weak discrepancy, where (ii) must hold not for the sum but for each individual pair of incomparable elements of *P*. We prove that, unlike the latter, the total weak and fractional total weak discrepancy of *P* are always the same, and we give a polynomial-time algorithm to find their common value. We use linear programming duality and complementary slackness to obtain this result.

#### Recommended Citation

Shuchat, A., Shull, R., & Trenk, A. N. (2011). The Total Weak Discrepancy of a Partially Ordered Set. Ars Mathematica Contemporanea 4: 95–109.

#### Version

Publisher's version

## Comments

Published in: Ars Mathematica Contemporanea 4: 95–109 (2011).

http://amc-journal.eu/index.php/amc/article/view/159