Improved bounds for scheduling conflicting jobs with minsum criteria

We consider a general class of scheduling problems where a set of conflicting jobs needs to be scheduled (preemptively or nonpreemptively) on a set of machines so as to minimize the weighted sum of completion times. The conflicts among jobs are formed as an arbitrary conflict graph. Building on the framework of Queyranne and Sviridenko [2002b], we present a general technique for reducing the weighted sum of completion-times problem to the classical makespan minimization problem. Using this technique, we improve the best-known results for scheduling conflicting jobs with the min-sum objective, 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 nonpreemptive scheduling on interval graphs.We also improve the results of Kim [2003] for scheduling jobs on line graphs and for resource-constrained scheduling.