Ranking the vertices of a complete multipartite paired comparison digraph

A paired comparison digraph (abbreviated to PCD) D = (V,A) is a weighted digraph in which the sum of the weights of arcs, if any, joining two distinct vertices equals one. A one-to-one mapping α from V onto {1,2,..., |V|} is called a ranking of D. For every ranking α, an arc vu ∈ A is said to be forward if α(v) < α(u), and backward otherwise. The length of an arc vu is l(vu) = ε(vu)|α(v) - α(u)|, where ε(vu) is the weight of vu. The forward (backward) length f _D(α) (b_D(α)) of α is the sum of the lengths of all forward (backward) arcs of D. A ranking α is forward (backward) optimal if f(α) is maximum ( b(α) is minimum). Kano (Discrete Appl. Math. 17 (1987) 245-253) characterized all backward optimal rankings of a complete multipartite PCD L and raised the problem to characterize all forward optimal rankings of a complete multipartite PCD L. We show how to transform the last problem into the single machine job sequencing problem of minimizing total weighted completion time subject to precedence "parallel chains" constraints. This provides an algorithm for generating all forward optimal rankings of L as well as a polynomial algorithm for finding the average rank of every vertex in L over all forward optimal rankings of L.