A Rogues’ Gallery of Reductions

Abstract

If one can solve problem A using a black box that solves problem B, one can reasonably say that problem A is “not much harder than” problem B, the degree of the “not much” being linked to how powerful and extensive the use of B is. Thus, reductions provide a means of classifying the relative hardness of sets. If A reduces to B and B reduces to A, then we can reasonably say that A and B are of “about the same” hardness. Of course, the closeness of the relationship between A and B will again depend on how powerful the reduction is. The more computationally weak the reduction is, the stronger the claim we can make about the similarity of hardness of A and B. Over the years, a rich collection of reductions has been developed to aid in classifying the relative hardness of sets. In this chapter, we define the key reductions, mention some standard notational shorthands, and then present some comments about reductions and their relative powers. We also discuss a centrally important reduction, Cook’s reduction, which is the reduction that proves that SAT is NP-complete.