TY - JOUR

T1 - Tools for multicoloring with applications to planar graphs and partial k-trees

AU - Halldórsson, Magnús M.

AU - Kortsarz, Guy

PY - 2002/2

Y1 - 2002/2

N2 - We study graphs multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary (preemptive schedule). We consider both the sum-of-completion times measure, or the sum of the last color assigned to each vertex, as well as the more common makespan measure, or the number of colors used. In this paper, we study two fundamental classes of graphs: planar graphs and partial k-trees. For both classes, we give a polynomial time approximation scheme (PTAS) for the multicoloring sum, for both the preemptive and nonpreemptive cases. On the other hand, we show the problem to be strongly NP-hard on planar graphs, even in the unweighted case, known as the sum coloring problem. For a nonpreemptive multicoloring sum of partial k-trees, we obtain a fully polynomial time approximation scheme. This is based on a pseudo-polynomial time algorithm that holds for a general class of cost functions. Finally, we give a PTAS for the makespan of a preemptive multicoloring of partial k-trees that uses only O(log n) preemptions. These results are based on several properties of multicolorings and tools for manipulating them, which may be of more general applicability.

AB - We study graphs multicoloring problems, motivated by the scheduling of dependent jobs on multiple machines. In multicoloring problems, vertices have lengths which determine the number of colors they must receive, and the desired coloring can be either contiguous (nonpreemptive schedule) or arbitrary (preemptive schedule). We consider both the sum-of-completion times measure, or the sum of the last color assigned to each vertex, as well as the more common makespan measure, or the number of colors used. In this paper, we study two fundamental classes of graphs: planar graphs and partial k-trees. For both classes, we give a polynomial time approximation scheme (PTAS) for the multicoloring sum, for both the preemptive and nonpreemptive cases. On the other hand, we show the problem to be strongly NP-hard on planar graphs, even in the unweighted case, known as the sum coloring problem. For a nonpreemptive multicoloring sum of partial k-trees, we obtain a fully polynomial time approximation scheme. This is based on a pseudo-polynomial time algorithm that holds for a general class of cost functions. Finally, we give a PTAS for the makespan of a preemptive multicoloring of partial k-trees that uses only O(log n) preemptions. These results are based on several properties of multicolorings and tools for manipulating them, which may be of more general applicability.

UR - http://www.scopus.com/inward/record.url?scp=0036488753&partnerID=8YFLogxK

U2 - 10.1006/jagm.2001.1210

DO - 10.1006/jagm.2001.1210

M3 - Article

AN - SCOPUS:0036488753

VL - 42

SP - 334

EP - 366

JO - Journal of Algorithms

JF - Journal of Algorithms

SN - 0196-6774

IS - 2

ER -