  We consider high dimensional variants of the maximum flow and minimum cut
problems in the setting of simplicial complexes and provide both algorithmic
and hardness results. By viewing flows and cuts topologically in terms of the
simplicial (co)boundary operator we can state these problems as linear programs
and show that they are dual to one another. Unlike graphs, complexes with
integral capacity constraints may have fractional max-flows. We show that
computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial
definition of a simplicial cut that seems more natural in the context of
optimization problems and show that computing such a cut is NP-hard. However,
we provide conditions on the simplicial complex for when the cut found by the
linear program is a combinatorial cut. For $d$-dimensional simplicial complexes
embedded into $\mathbb{R}^{d+1}$ we provide algorithms operating on the dual
graph: computing a maximum flow is dual to computing a shortest path and
computing a minimum cut is dual to computing a minimum cost circulation.
Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes,
prove its correctness, and provide a heuristic which guarantees it to halt.
