TY - JOUR

T1 - Improved bounds for sum multicoloring and scheduling dependent jobs with minsum criteria

AU - Gandhi, Rajiv

AU - Halldórsson, Magnús M.

AU - Kortsarz, Guy

AU - Shachnai, Hadas

PY - 2005

Y1 - 2005

N2 - We consider a general class of scheduling problems where a set of dependent jobs needs to be scheduled (preemptively or non-preemptively) on a set of machines so as to minimize the weighted sum of completion times. The dependencies among the jobs are formed as an arbitrary conflict graph. An input to our problems can be modeled as an instance of the sum multicoloring (SMC) problem: Given a graph and the number of colors required by each vertex, find a proper muticoloring which minimizes the sum over all vertices of the larges color assigned to each vertex. In the preemptive case (pSMC), each vertex can receive an arbitrary subset of colors; in the non-preemptive case (nsSMC), the colors assigned to each vertex need to be contiguous. SMC is known to be no easier than classic graph coloring, even in the case of unit color requirements. Building on the framework of Queyranne and Svirikenko (J. of Scheduling, 5:287-305, 2002), we present technique for reducing the sum multicoloring problem to classical graph multicoloring. Using the technique, we improve the best known results for pSMC and npSMC on several fundamental classes of graphs, including line graphs, (k+1)-claw free graphs and perfect graphs. In particular, we obtain the first constant factor approximation ratio for npSMC on interval graphs, on which our problems have numerous applications. We also improve the results of Kim (SODA 2003,97-98) for npSMC of line graphs and for resource-constrained scheduling.

AB - We consider a general class of scheduling problems where a set of dependent jobs needs to be scheduled (preemptively or non-preemptively) on a set of machines so as to minimize the weighted sum of completion times. The dependencies among the jobs are formed as an arbitrary conflict graph. An input to our problems can be modeled as an instance of the sum multicoloring (SMC) problem: Given a graph and the number of colors required by each vertex, find a proper muticoloring which minimizes the sum over all vertices of the larges color assigned to each vertex. In the preemptive case (pSMC), each vertex can receive an arbitrary subset of colors; in the non-preemptive case (nsSMC), the colors assigned to each vertex need to be contiguous. SMC is known to be no easier than classic graph coloring, even in the case of unit color requirements. Building on the framework of Queyranne and Svirikenko (J. of Scheduling, 5:287-305, 2002), we present technique for reducing the sum multicoloring problem to classical graph multicoloring. Using the technique, we improve the best known results for pSMC and npSMC on several fundamental classes of graphs, including line graphs, (k+1)-claw free graphs and perfect graphs. In particular, we obtain the first constant factor approximation ratio for npSMC on interval graphs, on which our problems have numerous applications. We also improve the results of Kim (SODA 2003,97-98) for npSMC of line graphs and for resource-constrained scheduling.

UR - http://www.scopus.com/inward/record.url?scp=23944453977&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-31833-0_8

DO - 10.1007/978-3-540-31833-0_8

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AN - SCOPUS:23944453977

SN - 0302-9743

VL - 3351

SP - 68

EP - 82

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

T2 - Second International Workshop on Approximation and Online Algorithms, WAOA 2004

Y2 - 14 September 2004 through 16 September 2004

ER -