Decomposing weighted digraphs into sums of chains

Abstract Suppose each of m individuals rank orders n items, denoted by 1,2,…,n. A summary of these rankings is provided by the function w that has wii = 0 for all i, and wij + wij = m for all distinct i and j, where wij is the number of individuals who rank i ahead of j. This paper examines the inverse problem of decomposing an aggregate function w into m rankings. Given a w that satisfies the preceeding conditions, the problem is to determine whether there are m rankings of {1,2,…,n} that have w as their summary and if so, to specify such rankings. It is shown that the problem has a ‘simple’ solution only if either m≦2 or n≦5. When n≦5, w can be decomposed into a sum of m rankings if and only if wij + wjk + wki≦2m for all i,j,k in {1,…,n}. Difficulties that arise for n≧6 and m≧3 are noted.