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?