Which NP-Complete problem has the fastest known algorithm?

Question

In terms of worst-case asymptotic runtime, which NP-complete problem has the fastest-known (exact) algorithm and what is the algorithm? Is there something known that is faster than $O(n^2*2^n)$?

• Guildenstern: What algorithm has running time $O(n^2 \cdot 2^n)$? EDIT: I assume you mean the Held–Karp algorithm for Traveling Salesman.
• Ainsley H.: You can take a look at the answers to the question Are there subexponential-time algorithms for NP-complete problems?.
• Raphael: "Faster than $O(\_)$" does not make sense. You mean $\Theta$? Or is the question, "Is there an algorithm with a better proven upper runtime bound than $O(\_)$?"
• Wuschelbeutel Kartoffelhuhn: The latter. It's valid point; there could be an algorithm A that's faster than B in practice but not with a tighter upper bound. I'm not sure why it doesn't make sense to say "faster than an upper bound" rather than "faster than a lower AND upper bound"...

Answer 1

Vertex Cover has an algorithm running in time 1ドル.2738^k + nk,ドル and is thus faster than 2ドル^n n^2,ドル even with $k=n$. You can check out Table of FPT races for a short list of FPT running times of different problems. Here, $n$ is the number of vertices and $k$ is the solution size.

Also, the question Are there subexponential-time algorithms for NP-complete problems? addresses similar questions.

• Raphael: The questions asks for fastest known algorithms and the table you link to does have "faster" algorithms than the VC one (in particular subexponential ones), so it's probably not the best to cite.
• Ainsley H.: See also this similar question and David Eppstein's answer Best-case Running-time to solve an NP-Complete problem on mathoverflow.
• Ainsley H.: @Raphael Yes, for instance Minimum Fill-In has an algorithm which for every $\epsilon > 0,ドル runs in $O( (1+\epsilon)^k + \text{poly}(n))$ time.