Minimal Controllability Problems

TL;DR

This paper studies the minimal controllability problem in linear systems, proving its NP-hardness and proposing an efficient greedy heuristic that performs well in practice for random graphs.

Contribution

It establishes the NP-hardness of the minimal controllability problem and introduces a simple greedy algorithm that effectively approximates the optimal solution.

Findings

NP-hardness of the problem proven

Greedy heuristic matches the inapproximability barrier

Heuristic performs well on random graph experiments

Abstract

Given a linear system, we consider the problem of finding a small set of variables to affect with an input so that the resulting system is controllable. We show that this problem is NP-hard; indeed, we show that even approximating the minimum number of variables that need to be affected within a multiplicative factor of $c \log n$ is NP-hard for some positive $c$. On the positive side, we show it is possible to find sets of variables matching this inapproximability barrier in polynomial time. This can be done by a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs demonstrate this heuristic almost always succeeds at findings the minimum number of variables.