Weak and strong completeness

Question

What does a pseudo-polynomial algorithm tell us about the problem it solves? I don't see how running time improves if the algorithm is exponential in the input length and polynomial in the input value; so how do we explain this shift from exponential to polynomial?

Answer 1

As it is stated in "Computers and Intractability: a Guide to the Theory of NP-Completeness" A pseudo-polynomial-time algorithm will display 'exponential behavior' only when confronted with instances containing 'exponentially large' numbers, which might be rare for the applications we are interested in. If so, this type of algorithm might serve our purposes almost as well as a polynomial time algorithm.”

You can consider knapsack as a good example of weakly-Np Complete problem. In this case the complexity of the dynamic programming solution is $O(nW)$ which is well in most practical cases.

It is known that there is no pseudo-polynomial time algorithm for Strong NP-Complete problems (like Steiner Tree) unless P=NP.

Answer 2

This answer pertains to quasipolynomial algorithms rather than pseudopolynomial ones.

A quasipolynomial algorithm tells us that the problem is probably not NP-hard. The (somewhat) commonly-believed Exponential Time Hypothesis (ETH) states that 3SAT on $n$ variables requires time $2^{\Omega(n)}$. Since 3SAT is in NP, the ETH implies that any NP-complete problem requires time $2^{n^{\Omega(1)}}$, which grows faster than quasipolynomial (assuming by the latter is meant $2^{\log^{O(1)} n}$).