Two decision problems are said to be polynomial-time isomorphic if there is a membership-preserving one-to-one correspondence between the instances of the two problems, which is both polynomial-time computable and polynomial-time invertible. Thus, this is a stronger notion than the notion of equivalence under the polynomial-time many-one reductions. It has, however, been observed that many reductions constructed between natural NP-complete problems are not only many-one reductions but can also be easily converted to polynomial-time isomorphisms. This observation leads to the question of whether two NP-complete problems must be polynomial-time isomorphic. Indeed, if this is the case then, within a polynomial factor, all NP-complete problems are actually identical and any heuristics for a single NP-complete problem works for all of them. In this chapter, we study this question and the related questions about the isomorphism of EXP-complete and P-complete problems.
Let us start with a simple example.