Abstract
This paper takes a new approach to studying the NP-isomorphism problem initiated by Berman and Hartmanis. We consider polynomial-time computable and invertible bijections among NP-complete sets that preserve the underlying natural distributions on random instances. We show that there are natural NP-complete problems which are not polynomially isomorphic when instances are chosen at random because no polynomial-time isomorphisms can preserve the underlying natural distributions on instances of these problems, On the other hand, we show that all the known average-case NP-complete problems under many-one reductions are polynomially isomorphic with respect to their distributions on random instances. Simple and accessible proofs to the known average-case NP-complete problems are presented when establishing the isomorphism result. New average-case NP-complete problems are also presented.
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