Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees

Magnús M. Halldórsson, Guy Kortsarz
2002 Journal of Algorithms  
We study graph 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 (non-preemptive 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
more » ... 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 non-preemptive 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 non-preemptive 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. * Preliminary version appears in APPROX '99 [HK99] .
doi:10.1006/jagm.2001.1210 fatcat:eyi2w2ayy5ebjh62sehoxr7h6y