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In the $k$-PATH problem, we receive as input a graph $G$ and an integer $k$. The goal is to decide whether there exists a simple path of length $k$ in $G$.

A $\alpha$-approximation for $k$-PATH is an algorithm which given an input $\langle G,k\rangle$ either confirms that a path of length $k/\alpha$ exists in $G$, or denies the existence of a $k$-path. It may act arbitrarily if the longest simple path in $G$ is at least of size $k/\alpha$ and at most of size $k$.

Is there a constant $\alpha>0$ for which $k$-PATH is $\alpha$-approximable in poly time?

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Karger, Motwani and Ramkumar (1997) discuss this question. They show that if any polynomial-time algorithm can approximate the longest path to a ratio of $2^{O(\log^{1−\epsilon} n)}$, for any $\epsilon>0$, then NP has a quasi-polynomial deterministic time simulation; this results even holds for graphs of bounded degree.

David R. Karger, Rajeev Motwani, G. D. S. Ramkumar:
On Approximating the Longest Path in a Graph.
Algorithmica 18(1): 82-98 (1997)

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  • $\begingroup$ My impression (from the one seeming clarification I found during my skim of that paper) was that their results are about finding a fairly long path, rather than just approximating the length of the longest paths. ​ ​ $\endgroup$ – user6973 Mar 14 '16 at 14:45
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    $\begingroup$ @RickyDemer The hardness reduction should work for the length as well. $\endgroup$ – Chandra Chekuri Mar 14 '16 at 18:25

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