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From: shor@research.att.com (Peter Shor)
Subject: Re: P!=NP does NOT imply NPC cannot be solved
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In article <41fn9u$121@chronicle.mti.sgi.com>, naylor@mti.sgi.com (William Clark Naylor) writes:
|> 
|> It seems to me that people discussing NP-completeness have been overlooking 
|> something very obvious and I would like to state it here as a reminder.
|> 
|> As an engineer, I am interested in solving NP-complete problems quickly 
|> by computer because this would give me the ability to solve many economically
|> important problems that I cannot now solve.  If somebody can find a practical
|> computer algorithm for solving some NP-complete problem, I can write 
|> a compiler/translator from any other NP-complete problem and use the algorithm
|> as a subroutine to solve the problem.  The P-time algorithm could be used
|> to solve a huge variety of problems, much as the simplex method is used
|> to solve a huge variety of problems today.


I think this is an excellent point, but there is one thing you have to be careful
about.  There are many NP-complete problems which seem relatively tractable
"on average," and others which are very difficult "on average."   Unfortunately, 
when you try to reduce one to the other, you generally end up blowing up the 
size of the problem and getting a non-average instance of it, making the 
resulting problem just as hard to solve as the original, so that the relatively 
easy NP-complete problems don't help in solving the hard NP-complete problems.

Thus, a good average-case solver for satisfiability may not help at all in finding
chromatic number.  The remarkable thing about linear programming is that ALL the
linear programs people come up with in practice seem to behave well with the
simplex method (some much better than others, admittedly, but all behave a lot 
better than the worst-case examples).


Peter Shor
