The Complexity of Tree Multicolorings

The multicoloring problem is that given a graph G and integer demands x(v) for every vertex v, assign a set of x(v) colors to vertex v, such that neighboring vertices have disjoint sets of colors. In the preemptive sum multicoloring problem the finish time of a vertex is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The study of this problem is motivated by applications in scheduling. Answering a question of Halldórsson et al. [4], we show that the problem is strongly NP-hard in binary trees. As a first step toward this result we prove that list multicoloring of binary trees is NP-complete.