Improved Bounds for Sum Multicoloring and Scheduling Dependent Jobs with Minsum Criteria

We consider a general class of scheduling problems where a set of dependent jobs needs to be scheduled (preemptively or nonpreemptively) 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 multicoloring which minimizes the sum over all vertices of the largest color assigned to each vertex. In the preemptive case (pSMC), each vertex can receive an arbitrary subset of colors; in the non-preemptive case (npSMC), 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 Sviridenko (J. of Scheduling, 5:287-305, 2002 ), we present a general 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 resourceconstrained scheduling.