The total linear discrepancy of an ordered set

In this paper we introduce the notion of the total linear discrepancy of a poset as a way of measuring the fairness of linear extensions. If L is a linear extension of a poset P , and x , y is an incomparable pair in P , the height difference between x and y in L is | L ( x ) − L ( y ) | . The total linear discrepancy of P in L is the sum over all incomparable pairs of these height differences. The total linear discrepancy of P is the minimum of this sum taken over all linear extensions L of P . While the problem of computing the (ordinary) linear discrepancy of a poset is NP-complete, the total linear discrepancy can be computed in polynomial time. Indeed, in this paper, we characterize those linear extensions that are optimal for total linear discrepancy. The characterization provides an easy way to count the number of optimal linear extensions.