Isomorphism and NP-completeness

Abstract

An interesting area of research in Complexity Theory is the study of the structural properties of the complete problems in NP. The analogy with the class of the recursively enumerable sets, provided by the polynomial time hierarchy, suggests that they might have properties similar to those of the r.e.-complete sets, like being pairwise isomorphic, in the appropriate sense; in fact, sufficient conditions are known for certain sets being isomorphic under polynomial time computable, polynomially invertible bijections, and no NP-complete set has been proved to be non-isomorphic to SAT in this sense. It was conjectured that all NP-complete sets are polynomially isomorphic, a statement which is known as the BermanHartmanis conjecture. Several consequences follow from this conjecture; among them, of course, P ≠ NP.