  With the rising importance of large-scale network control, the problem of
actuator placement has received increasing attention. Our goal in this paper is
to find a set of actuators minimizing the metric that measures the average
energy consumption of the control inputs while ensuring structural
controllability of the network. As this problem is intractable, greedy
algorithm can be used to obtain an approximate solution. To provide a
performance guarantee for this approach, we first define the submodularity
ratio for the metric under consideration and then reformulate the structural
controllability constraint as a matroid constraint. This shows that the problem
under study can be characterized by a matroid optimization involving a weakly
submodular objective function. Then, we derive a novel performance guarantee
for the greedy algorithm applied to this class of optimization problems.
Finally, we show that the matroid feasibility check for the greedy algorithm
can be cast as a maximum matching problem in a certain auxiliary bipartite
graph related to the network graph.
