Batch Coloring Tree-like Graphs

Batch scheduling of conflicting jobs is modeled by batch coloring of a graph. Given an undirected graph and the number of colors required by each vertex, we need to find a proper batch coloring of the graph, i.e., partition the vertices to batches which are independent sets, and to assign to each batch a contiguous set of colors, whose size equals to the maximum color requirement of any vertex in this batch. When the objective is to minimize the sum of job completion times, we get the batch sum coloring problem; when we want to minimize the maximum completion time of any job (or, the makespan) we get the max coloring problem. Given the hardness of batch coloring on general graphs, already for the special case of unit length jobs (our problems then reduce to sum coloring and the classic graph coloring problem, respectively), it is natural to seek out classes of graphs where effective solutions can be obtained efficiently. In this paper we give the first polynomial time approximation schemes for batch sum coloring on tree-like graphs, as well as for planar graphs. For the max-coloring problem, we improve upon previous results for several classes of tree-like graphs, as well as for perfect graphs.