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.

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