  The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph $D=(V,A)$ with nonnegative integral
weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$,
decide whether $D$ has a closed directed walk containing every arc of $R$ and
of total weight at most $\ell$. Let $k$ be the number of weakly connected
components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$,
i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a
function of $k$ only and the $O^*$ notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
$O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in
$D$. We also show that the same result holds for the undirected version of
DRPP, where $D$ is a connected undirected multigraph.
