Minimum sum multicoloring on the edges of trees

The edge multicoloring problem is that given a graph G and integer demands x(e) for every edge e, assign a set of x(e) colors to edge e, such that adjacent edges have disjoint sets of colors. In the minimum sum edge multicoloring problem the finish time of an edge is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The main result of the paper is a polynomial time approximation scheme for minimum sum multicoloring the edges of trees. We also show that the problem is strongly NP-hard for trees, even if every demand is at most 2.