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 ofD. 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 `(vu) = (vw)|α(v)− α(u)|, where (vw) is the weight of vu. The forward ( backward) length fD(α) (bD(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). M. Kano (Disc. Appl. Math., 17 (1987) 245-253) characterized all backward optimal rankings of a complete multipartite PCD D 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 ∗Corresponding author. This work was supported by the Danish Research Council under grant no. 11-0534-1. The support is gratefully acknowledged.