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From: paulv@boring.UUCP
Newsgroups: net.math
Subject: Re: Karmarkar algorithm
Message-ID: <6285@boring.UUCP>
Date: Tue, 15-Jan-85 14:32:58 EST
Article-I.D.: boring.6285
Posted: Tue Jan 15 14:32:58 1985
Date-Received: Thu, 17-Jan-85 14:04:55 EST
References: <7700001@hplvle.UUCP>
Reply-To: paulv@boring.UUCP (Paul Vitanyi)
Organization: CWI, Amsterdam
Lines: 47
Apparently-To: rnews@mcvax.LOCAL

In article <7700001@hplvle.UUCP> drick@hplvle.UUCP (drick) writes:
>[bug food]
>0. Karmarkar's paper has not been published yet.  
>1.  "A direct comparison of cpu times shows that the new method beats
>MPSX/370 - a commercial implementation of the simplex method - by more
>than a factor of 50 on problems with several thousand variables.  
>4.  The existence of this algorithm reestablishes Turing's model of
>computation.  Briefly, Turing said a "good" algorithm is a polynomial-
>time algorithm.  The only previously known polynomial-time algorithm
>for linear programming, the ellipsoid method, was slower in practice
>than the simplex method (an exponential-time method), thus casting
>doubt on Turing's conjecture.
>David L. Rick
>Loveland Instrument Division
>Hewlett-Packard Company
>hplabs!hplvla!hplvle

0. Karmarkar's method appears in the  Proceedings of the 
16th ACM Symposium on Theory of Computing, Washington D.C., April-May 1984,
pp 302-311 (ACM Order #508840).
1. It is claimed to have beaten the blah package for some specific
problems, not for all problems. Maybe K's Algorithm
runs better only on this eclectic set.
4. The algorithm has not more to do with T's model of computation than
any other algorithm. Turing had been dead for over a decade (and his
paper published for thirty years) before somebody (Cobham-Hartmanis-
Stearns-Cook-Karp '65~'70s) started talking about polynomial time
algorithms. The simplexmethod runs in linear time for most
*real-life* problems, but can be forced to spend exponential time
on some *cooked-up* problems. It was unknown whether LP was as
difficult as a large class of problems (NP-Complete Problems) for
which only exponential *worst-case* time algorithms are known. When
Khachians Ellipsoid Algorithm came along (1979) it was shown
that LP is not difficult, in that sense, i.e., the latter algorithm
has a polynomial worst-case running time. For *practical* use
the simplexmethod appeared to perform better. The Karmarkar
algorithm is *claimed* to perform not only better than any other
in the *theoretically* interesting *worst-case*, but also in
*practical use*. About the latter claim there is no consensus at all;
a lot of investigation is going on. The worst-case running time
of K's algorithm is worse than what is considered acceptable for
a common run of Simplex; the question is whether it in general
performs better on common natural problems than does Simplex.
	Nothing in this story throws, or has thrown, any doubt
on Turing's or anybody elses conjectures. It has no bearing
on the value of any model of computation.