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I came across this question while studying for an exam:

T/F: Suppose we can show for some fixed $k$, an NP-complete problem P has a time $O(n^k)$ algorithm. Then every problem in NP has a $O(n^k)$ time algorithm.

I think the answer is false, since we can't reduce NP-complete to NP-hard in linear time, right? Or am I completely misunderstanding reductions/NP problems? Any help would be greatly appreciated. Thank you.

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Yes, the answer the the question is "false".

By contradiction, assume every problem in $NP$ has a $O(n^k)$ algorithm, for some fixed $k$. That is, $NP \subseteq {\sf DTIME}(O(n^k))$

This would contradict the time hierarchy theorem, which implies the strict inclusion

$$ {\sf DTIME}(O(n^k)) \subset {\sf DTIME}(O(n^{2k})) $$

However, we also have

$$ {\sf DTIME}(O(n^{2k})) \subseteq NP \subseteq {\sf DTIME}(O(n^{k})) $$

which contradicts the previous strict inclusion.

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If $\mathbf{P} = \mathbf{NP}$, can all $\mathbf{NP}$ problems be solved within time $O(n^k)$ for fixed $k$?

No, because the time hierarchy theorem says that there are problems that can be solved in time $O(n^{k+1})$ that can't be solved in time $O(n^k)$.

we can't reduce NP-complete to NP-hard in linear time

Any $\mathbf{NP}$-complete problem is $\mathbf{NP}$-hard by definition.

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  • $\begingroup$ Note that the title is my attempt at creating a more snappy yet accurate one than the OP's. For that, I assumed that P = NP as a consequence of the problem statement was obvious. $\endgroup$ – Raphael Nov 5 '18 at 14:37
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    $\begingroup$ @Raphael OK. I still think my answer is appropriate to the question body. $\endgroup$ – David Richerby Nov 5 '18 at 14:38

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