Sum Multicoloring of Graphs

Amotz Bar-Noy
Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel.

Magnús M. Halldórsson
Science Institute, University of Iceland, IS-107 Reykjavik, Iceland.
AND University of Bergen

Guy Kortsarz
Computer Science Dept., Open University, Ramat Aviv, Israel.

Ravit Salman
Dept. of Mathematics, Technion, Haifa 32000, Israel.

Hadas Shachnai
The Department of Computer Science, Technion, Haifa 32000, Israel.

Scheduling dependent jobs on multiple machines is modeled by the graph multicoloring problem. In this paper we consider the problem of minimizing the average completion time of all jobs. This is formalized as the sum multicoloring problem: Given a graph and the number of colors required by each vertex, find a multicoloring which minimizes the sum of the largest colors assigned to the vertices. It reduces to the known sum coloring problem when each 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-scheduling 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 models, via a link to the sum coloring problem. Thus, for classes of graphs for which the independent set problem is r-approximable, we obtain O(r)-approximations for preemptive and co-scheduling sum multicoloring, and O(r log n)-approximation for nonpreemptive sum multicoloring. In addition, we give constant ratio approximations for a number of fundamental classes of graphs, including bipartite, line, bounded degree, and planar graphs.

Submitted, July 3, 1999.