The [formula omitted]-discrepancy of a poset

Abstract

Linear discrepancy and weak discrepancy have been studied as a measure of fairness in giving integer ranks to the points of a poset. In linear discrepancy, the points are totally ordered, while in weak discrepancy, ties in rank are permitted. In this paper we study the t -discrepancy of a poset, which can be viewed as a hybrid between linear and weak discrepancy, in which at most t points can receive the same rank. Interestingly, t -discrepancy is not a comparability invariant while both linear and weak discrepancy are. We show that for a poset P and positive integers t and k , the decision problem of determining whether the t -discrepancy of P is at most k is NP-complete in general; however, we give a polynomial time algorithm for computing the t -discrepancy of a semiorder. We also find the t -discrepancy for posets that are the disjoint sum of chains.