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From: sbloch@adl15.adelphi.edu (Stephen Bloch)
Subject: Re: NP-complete and exponential time
Message-ID: <CzMHFA.991@adl33cc.adelphi.edu>
Keywords: P; NP; NP-complete; exponential time; sub-exponential time
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Date: Mon, 21 Nov 1994 14:54:45 GMT
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Richard Pinch <rgep@emu.pmms.cam.ac.uk> wrote:
>In a discussion over on comp.ai.genetic, a question arose as to whether
>a subexponential (but superpolynomial) algorithm for an NP-complete problem 
>would have any implications for the P = NP problem.  
>
>For a concrete example, what could one deduce from a time estimate of 
>exp(C n^1/2 (log n)^1/2) on n bits of input: this sort of function occurs in 
>number-theoretic problems such as factoring and discrete logarithm.

That doesn't look subexponential to me.  When I think of "subexponential
time", I usually think "dominated by exp(p(n)) for all nonconstant
polynomials p", even polynomials with degree less than one.  For
example, exp((log n)^2) qualifies as subexponential.  Anyway, if an
NP-complete problem is in DTIME(F(n)), where F is a class of functions
closed under composition with a polynomial (e.g. exp(Cn^1/2(logn)^1/2)),
it implies that NP is contained in DTIME(F(n)).  It decidedly does NOT
imply P=NP; I went to a good seminar on this last year (Judy, are you
listening?)

--
                                                 Stephen Bloch
                                           sbloch@boethius.adelphi.edu
                                        Math/CS Dept, Adelphi University
