Suppose problem is hard to approximate, and an L-reduction translates its instance into an instance of . We are testing what a hypothetical approximation algorithm for would give us. That algorithm supplies a target solution of value ; the reduction converts it back into a source solution of value . For maximisation, write for the optima. The reduction guarantees
Take , , , . If the target algorithm returns , it is a approximation and misses points. Conversion misses at most source points, so : at least a source approximation. The lower bar shows the worst allowed source outcome; conversion may do better. The diagram tracks points missing from the optimum, rather than treating and as equal scores.
Why hardness moves forwards
In general, a target ratio means . The two reduction bounds turn that into a source guarantee, step by step:
Thus a target algorithm would produce a source algorithm. If the source cannot be approximated beyond a ratio , neither can the target beyond : a strictly better target ratio would make the source ratio strictly better than . This is inapproximability transfer. The direction reverses because the hypothetical algorithm runs on the target, while the contradiction is known for the source.