We consider a general class of scheduling problems where a set of conflicting 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 conflicts among the jobs are formed as an arbitrary conflict graph. Building on the framework of Queyranne and Sviridenko (J. of Scheduling, 5:287-305, 2002 ), we present a general technique for reducing the weighted sum of completion times problem to the classical makespan

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... mization problem. Using this technique, we improve the best known results for scheduling conflicting jobs with minsum 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 non-preemptive scheduling on interval graphs. We also improve the results of Kim (SODA 2003, 97-98 ) for scheduling jobs on line graphs and for resource-constrained scheduling.