An asymmetric scheduling instance differs from an atomic selfish routing instance in the following two respects. First, the underlying network is restricted to a common source vertex s, a common sink vertex t, and a set of parallel links that connect s to t. On the other hand, we allow different players to possess different strategy sets: each player i has a prescribed subset Si of the links that it is permitted to use.
Show that every asymmetric scheduling instance is equivalent to an atomic selfish routing game. Your reduction should make use only of the cost functions of the original scheduling instance, plus possibly the all-zero cost function.