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The Isomorphism Conjecture for NP Manindra Agrawal

Summary: The Isomorphism Conjecture for NP
Manindra Agrawal
December 19, 2009
In this article, we survey the arguments and known results for and against the Isomorphism
1 Introduction
The Isomorphism Conjecture for the class NP states that all polynomial-time many-one complete
sets for NP are polynomial-time isomorphic to each other. It was made by Berman and Hart-
manis [21]1, inspired in part by a corresponding result in computability theory for computably
enumerable sets [50], and in part by the observation that all the existing NP-complete sets known
at the time were indeed polynomial-time isomorphic to each other. This conjecture has attracted
a lot of attention because it predicts a very strong structure of the class of NP-complete sets, one
of the fundamental classes in complexity theory.
After an initial period in which it was believed to be true, Joseph and Young [40] raised serious
doubts against the conjecture based on the notion of one-way functions. This was followed by
investigation of the conjecture in relativized worlds [33, 46, 27] which, on the whole, also suggested
that the conjecture may be false. However, disproving the conjecture using one-way functions, or
proving it, remained very hard (either implies P = NP). Hence research progressed in three distinct
directions from here.


Source: Agrawal, Manindra - Department of Computer Science and Engineering, Indian Institute of Technology Kanpur


Collections: Computer Technologies and Information Sciences