[miloshh]

Yeah, the common complaint against "P=NP would break cryptography" is "but what if the polynomial algorithm for (say) SAT is n^10000?". That is not very likely, in my opinion. Even though "polynomial" is not equivalent to "efficient", in practice all problems that have a polynomial algorithm also have an efficient algorithm.

[dfox]

There are many problems that have polynomial algorithm, that is not especially fast for practical problems (matrix multiplication comes to mind for example, for which there isn't even any proof that fastest known algorithm is in fact optimal)

[miloshh]

That's not a very good example - square matrix multiplication is O(n^1.5) (where n is the input size) and extremely fast in practice.

(Though, as you say, there could still be an O(n) algorithm - this is not known).

What I said is true even for linear programming (a P-complete problem) - it has polynomial and efficient algorithms - though the latter are actually not polynomial in the worst case. :)

[slug]

Can you give a reference for a matrix multiplication algorithm which is O(n^1.5) ? thanks.

[miloshh]

Note that I defined n as the input size...