Sum Multicoloring of Graphs

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 vert...

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Bibliographic Details
Published in:Journal of algorithms Vol. 37; no. 2; pp. 422 - 450
Main Authors: Bar-Noy, Amotz, Halldórsson, Magnús M., Kortsarz, Guy, Salman, Ravit, Shachnai, Hadas
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 01.11.2000
Elsevier
Subjects:
Applied sciences
chromatic sums
Combinatorics
Combinatorics. Ordered structures
dependent jobs
dining philosophers
Exact sciences and technology
graph coloring
Graph theory
Mathematics
multicoloring
Operational research and scientific management
Operational research. Management science
scheduling
Scheduling, sequencing
Sciences and techniques of general use
sum coloring
chromatic sums
scheduling
dining philosophers
graph coloring
multicoloring
dependent jobs
sum coloring
Starvation
Operations research
Dining philosopher
Scheduling
Graph theory
Multicoloring
Management
Optimization
Dependence
Graph colouring
Chromatic graph
dependent job
Chromatic sum
ISSN:0196-6774, 1090-2678
Online Access:Get full text
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Summary: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 ρ-approximable, we obtain O(ρ)-approximations for preemptive and co-scheduling sum multicoloring and O(ρlogn)-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.
ISSN:0196-6774
1090-2678
DOI:10.1006/jagm.2000.1106