Consider the following language: $$ L = \{ G | G \text{ has a VC of size at most } \log n \} $$

Does $L\in P$ or $L\in NPC$?

  • 1
    $\begingroup$ What do you about algorithms for vertex cover? $\endgroup$
    – Juho
    Commented Apr 25, 2019 at 8:07
  • $\begingroup$ @Juho I know $VC \in NPC$ obviously (verified easily). I also know that the language of graphs that has $VC$ of size at most $|V|/2$ is also in $NPC$ (proved using reduction from original $VC$ problem). But the same trick doesn't scale for $\log n$. $\endgroup$
    – galah92
    Commented Apr 25, 2019 at 8:11
  • $\begingroup$ I think this problem is duplicate of cs.stackexchange.com/questions/99710/… . $\endgroup$ Commented Apr 25, 2019 at 8:44
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    $\begingroup$ en.wikipedia.org/wiki/Vertex_cover#Fixed-parameter_tractability has enough information to figure this out simply. $\endgroup$ Commented Apr 25, 2019 at 8:50
  • $\begingroup$ @LukeMathieson "Bounded Search Tree" does work, thanks $\endgroup$
    – galah92
    Commented Apr 25, 2019 at 15:32

1 Answer 1


If you solve Vertex Cover with the simple branching algorithm, you achieve an FPT running time of $2^k \text{poly}(n)$, where $k$ is an upper bound for the solution you are looking for.

Substituting $k$ for $\log n$ gives you the answer you are looking for.

  • $\begingroup$ Doesn't your answer assume that one cannot encode in $\text{poly}(k)$ a graph with $n=2^k$ vertices, while only the first $k$ vertices are not isolated? (See my answer) $\endgroup$ Commented Jun 8, 2019 at 10:16
  • $\begingroup$ Thank you for helping me to realize the convention is to assume that the length of the encoding of a graph is in $\Omega(|V|)$. I deleted my answer. (If you think I should undelete it, please say so.) $\endgroup$ Commented Jun 9, 2019 at 5:23

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