Complexity of the Minimum Input Selection Problem for Structural Controllability

We consider the minimum input selection problem for structural controllability (MISSC), stated as follows: Given a linear system ẋ = Ax, where A is a n × n state matrix with m nonzero entries, find the minimum number of states that need to be driven by an external input so that the resulting system is structurally controllable. The fastest algorithm to solve this problem was recently proposed by Olshevsky in (Olshevsky, 2015) and runs in Θ (m √ n) operations. In this paper, we propose an alternative algorithm to solve MISSC in min{O (m √ n) , Õ ( n ) , Õ ( m ) } operations. This running time is obtained by (i) proving that MISSC problem is computationally equivalent to the maximum bipartite matching (MBM) problem and (ii) considering the three fastest algorithms currently available to solve MBM, namely, the Hopcraft-Karp algorithm, the Mucha-Sankowski algorithm, and Madry’s algorithm. Furthermore, our algorithm can directly benefit from future improvements in MBM computation. Conversely, we also show that any algorithmic improvement on solving MISSC would result in an improvement in MBM computation, which would be of great interest for theoretical computer scientists.