  We analyze integer linear programs which we obtain after discretizing
two-dimensional subproblems arising from a trust-region algorithm for mixed
integer optimal control problems with total variation regularization. We
discuss NP-hardness of the discretized problems and the connection to
graph-based problems. We show that the underlying polyhedron exhibits
structural restrictions in its vertices with regards to which variables can
attain fractional values at the same time. Based on this property, we derive
cutting planes by employing a relation to shortest-path and minimum bisection
problems. We propose a branching rule and a primal heuristic which improves
previously found feasible points. We validate the proposed tools with a
numerical benchmark in a standard integer programming solver. We observe a
significant speedup for medium-sized problems. Our results give hints for
scaling towards larger instances in the future.
