Question

I am reading about P , NP and NP-Complete problems theory. Here is text snippet.

The class NP includes all problems that have polynomial-time
solutions, since obviously the solution provides a check. One would
expect that since it is so much easier to check an answer than to
come up with one from scratch, there would be problems in NP that do
not have polynomial-time solutions. To date no such problem has been
found, so it is entirely possible, though not considered likely by
experts, that nondeterminism is not such an important improvement. The
problem is that proving exponential lower bounds is an extremely
difficult task. The information theory bound technique, which we used
to show that sorting requires (n log n) comparisons, does not seem to
be adequate for the task, because the decision trees are not nearly
large enough.

My question is what does author mean by

1. by statement “To date no such problem has been found, so it is entirely possible, though
   not considered likely by experts, that nondeterminism is not such an important improvement.” ?

2. Another question what does author mean by in last statement by “because the decision trees are not nearly large enough.” ?

Thanks!

Answer 1

(1) I think the author means that no NP problem has been found, for which it is proven that it is not in P. Certainly there are problems in NP for which no polynomial solution is known, but that’s not the same as knowing that none exists.

If in fact P = NP (that is to say, if in fact there are no NP problems that don’t have a polynomial solution), then in some sense a nondeterministic machine is no “more powerful” than a deterministic machine, since they solve the same problems in polynomial time. Then we’d say “nondeterminism is not such an important improvement”.

(2) The way that the n log n proof works is that there are n! possible outputs from a sorting function, any one of which might be the correct one according to what order the input was in. Each comparison adds a two-legged branch to the tree of all possible states that a given comparison sort algorithm can get into. In order to sort any input, this “decision tree” must have enough branches to produce any of the n! possible re-orderings of the input, and hence there must be at least log(n!) comparisons. So, the lower bound on runtime comes from the size of the tree.

The author is saying that there are no known NP problems for which we’ve proved they require a tree so large that it implies a lower bound that is super-polynomial. Any such proof would prove P != NP.