Sum coloring interval and κ-claw free graphs with application to scheduling dependent jobs

We consider the sum coloring and sum multicoloring problems on several fundamental classes of graphs, including the classes of interval and κ-claw free graphs. We give an algorithm that approximates sum coloring within a factor of 1.796, for any graph in which the maximum κ-colorable subgraph problem is polynomially solvable. In particular, this improves on the previous best known ratio of 2 for interval graphs. We introduce a new measure of coloring, robust throughput, that indicates how "quickly" the graph is colored, and show that our algorithm approximates this measure within a factor of 1.4575. In addition, we study the contiguous (or non-preemptive) sum multicoloring problem on κ-claw free graphs. This models, for example, the scheduling of dependent jobs on multiple dedicated machines, where each job requires the exclusive use of a most κ machines. Assuming that κ is a fixed constant, we obtain the first constant factor approximation for the problem.