  The circuits of a polyhedron are a superset of its edge directions. Circuit
walks, a sequence of steps along circuits, generalize edge walks and are
"short" if they have few steps or small total length. Both interpretations of
short are relevant to the theory and application of linear programming.
  We study the hardness of several problems relating to the construction of
short circuit walks. We establish that for a pair of vertices of a
$0/1$-network-flow polytope, it is NP-complete to determine the length of a
shortest circuit walk, even if we add the requirement that the walk must be
sign-compatible. Our results also imply that determining the minimal number of
circuits needed for a sign-compatible decomposition is NP-complete. Further, we
show that it is NP-complete to determine the smallest total length (for
$p$-norms $\lVert \cdot \rVert_p$, $1 < p \leq \infty$) of a circuit walk
between a pair of vertices. One method to construct a short circuit walk is to
pick up a correct facet at each step, which generalizes a non-revisiting walk.
We prove that it is NP-complete to determine if there is a circuit direction
that picks up a correct facet; in contrast, this problem can be solved in
polynomial time for TU polyhedra.
