TY - JOUR

T1 - Sum Multicoloring of Graphs

AU - Bar-Noy, Amotz

AU - Halldórsson, Magnús M.

AU - Kortsarz, Guy

AU - Salman, Ravit

AU - Shachnai, Hadas

PY - 2000/11

Y1 - 2000/11

N2 - 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 p-approximable, we obtain O(ρ)-approximations for preemptive and co-scheduling sum multicoloring and O(ρ 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.

AB - 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 p-approximable, we obtain O(ρ)-approximations for preemptive and co-scheduling sum multicoloring and O(ρ 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.

KW - Chromatic sums

KW - Dependent jobs

KW - Dining philosophers

KW - Graph coloring

KW - Multicoloring

KW - Scheduling

KW - Sum coloring

UR - http://www.scopus.com/inward/record.url?scp=0013152316&partnerID=8YFLogxK

U2 - 10.1006/jagm.2000.1106

DO - 10.1006/jagm.2000.1106

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AN - SCOPUS:0013152316

SN - 0196-6774

VL - 37

SP - 422

EP - 450

JO - Journal of Algorithms

JF - Journal of Algorithms

IS - 2

ER -