Introducing a gap makes an NP-hard yes/no distinction visible in the optimum of one problem. A gap-preserving reduction takes an instance of that problem and constructs an instance of another. It need not preserve optimum values or even whether we minimise or maximise. It must preserve the separation: for example, a source minimisation optimum at most maps to a target maximisation optimum at least , while a source optimum greater than maps to a target optimum below . The same polynomial-time map works in both cases without knowing which case belongs to; source optima between and carry no promise.
Why is the target gap versus useful? If a hypothetical -approximation returns a feasible target value , then
Testing would distinguish the hard source cases. Introduction creates the hard gap; preservation moves it to a new problem. These numbers illustrate the required promises, not a particular reduction.