Sum multicoloring of graphs

Scheduling dependent jobs on multiple machines is modeled by the graph mult icoloring problem. In this paper we consider the problem of minimizing the average completion time of all jobs. This is formalized as the sum multicol oring problem: Given a graph and the number of colors required by each vert ex, find a multicoloring which minimizes the sum of the largest colors assi gned to the vertices. It reduces to the known sum coloring problem when eac h vertex requires exactly one color. This paper reports a comprehensive study of the sum multicoloring problem, treating three models: with and without preemptions, as well as co-scheduli ng where jobs cannot start while others are running. We establish a linear relation between the approximability of the maximum independent set problem and the approximability of the sum multicoloring problem in all three mode ls, via a link to the sum coloring problem. Thus, for classes of graphs for which the independent set problem is rho -approximable, we obtain O(rho)-a pproximations for preemptive and co-scheduling sum multicoloring and O(rho log n)-approximation for nonpreemptive sum multicoloring. In addition, we g ive constant ratio approximations for a number of fundamental classes of gr aphs, including bipartite, line, bounded degree, and planar graphs. (C) 200 0 Academic Press.