Inapproximability of hypergraph vertex cover and applications to scheduling problems

Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2¿-¿e for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2¿-¿e for minimizing the makespan in the assembly line problem. These results follow from a new inapproximability result for the Vertex Cover problem on k-uniform hypergraphs that is stronger and simpler than previous results. We show that assuming the UGC, for every k¿=¿2, the problem is inapproximable within k¿-¿e even when the hypergraph is almost k -partite