Sum multi-coloring of graphs

Scheduling dependent jobs on multiple machines is modeled by the graph multi-coloring problem. In this paper we consider the problem of minimizing the average completion time of all jobs. This is formalized as the sum multi-coloring (SMC) problem: Given a graph and the number of colors required by each vertex, find a multi-coloring which minimizes the sum of the largest colors assigned to the vertices. It reduces to the known sum coloring (SC) problem in the special case of unit execution times. This paper reports a comprehensive study of the SMC problem, treating three models: with and without preemption allowed, as well as co-scheduling where tasks cannot start while others are running. We establish a linear relation between the approximability of the maximum independent set (IS) and SMC in all three models, via a link to the SC problem. Thus, for classes of graphs where IS is p-approximable, we obtain O (p) -approximations for preemptive and co-scheduling SMC, and O(p • log n) for non-preemptive SMC. In addition, we give constant-approximation algorithms for SMC under different models, on a number of fundamental classes of graphs, including bipartite, line, bounded degree, and planar graphs.