An $r$-simple $k$-path is a {path} in the graph of length $k$ that
passes through each vertex at most $r$ times. The \simpath{r}{k}
problem, given a graph $G$ as input, asks whether there exists an
$r$-simple $k$-path in $G$. We first show that this problem is
NP-Complete. We then show that there is a graph $G$ that contains
an $r$-simple $k$-path and no simple path of length greater than
$4\log k/\log r$. So this, in a sense, motivates this problem
especially when one's goal is to find a short path that visits many
vertices in the graph while bounding the number of visits at each
vertex.

We then give a randomized algorithm that runs in
time
$$\poly(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the \simpath{r}{k}
on a graph with $n$ vertices with one-sided error. We also show
that a randomized algorithm with running time $\poly(n)\cdot
2^{(c/2)k/ r}$ with $c<1$ gives a randomized
algorithm with running time $\poly(n)\cdot 2^{cn}$ for the
Hamiltonian path problem in a directed graph - an outstanding open problem.
So in a sense our algorithm is optimal up to an $O(\log r)$
factor.