Batch Coloring Flat Graphs and Thin

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, namely, to partition the vertices to batches which are independent sets and assign to each batch a contiguous set of colors, whose size is equal 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 color requirements (known as 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 several classes of “non-thick” graphs that arise in applications. This includes paths, trees, partial k-trees, and planar graphs. Also, we give an improved O(n log n) exact algorithm for the max-coloring problem on paths.