Generalized edge-rankings of trees

In this paper we newly define a generalized edgeranking of a graph G as follows: for a positive integer c, a c-edgeranking of G is a labeling (ranking) of the edges of G with integers such that, for any label i, deletion of all edges with labels > i leaves connected components, each having at most c edges with label i. The problem of finding an optimal c-edge-ranking of G, that is, a c-edge-ranking using the minimum number of ranks, has applications in scheduling the manufacture of complex multipart products; it is equivalent to finding a c-edge-separator tree of G having the minimum height. We present an algorithm to find an optimal c-edge-ranking of a given tree T for any positive integer c in time O(n2 log A), where n is the number of vertices in T and A is the maximum vertex-degree of T. Our algorithm is faster than the best algorithm known for the case c= 1.