The t-discrepancy of a poset

Linear discrepancy and weak discrepancy have been studied as a measure of fairness in giving integer ranks to 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, 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 computing the tdiscrepancy of a poset is NP-complete in general but give a polynomial time algorithm for computing the t-discrepancy of a semiorder. We also find the t-discrepancy for posets that are the sum of chains and for the standard example of an n-dimensional poset.