Approximation Preserving Reductions

Abstract

WE HAVE seen in the preceding chapters that, even though the decision problems corresponding to most Np-hard optimization problems are polynomial-time Karp-reducible to each other, the optimization problems do not share the same approximability properties. The main reason of this fact is that Karp-reductions not always preserve the measure function and, even if this happens, they rarely preserve the quality of the solutions. It is then clear that a stronger kind of reducibility has to be used that not only maps instances of a problem P ₁ to instances of a problem P ₂, but it also maps back good solutions for P ₂ to good solutions for P ₁.