Minimal Controllability Problems (1304.3071v7)

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.

• The paper proves that finding the minimal set of variables to control a linear system is NP-hard and difficult to approximate.
• It proposes a polynomial-time greedy algorithm that provides a logarithmic approximation guarantee for finding such minimal control sets.
• The findings have practical implications for controlling large-scale networks in fields like biochemical systems, smart grids, and traffic.

Overview of "Minimal Controllability Problems"

This paper addresses the problem of minimal controllability of linear systems, where the goal is to find the smallest subset of system variables that need to be influenced by external inputs to achieve controllability. Specifically, the problem involves identifying these variables in such a way that the system becomes controllable, or equivalently, to affect the fewest possible number of variables with an input. The problem is shown to be computationally hard to approximate within a multiplicative factor of $c \log n$, suggesting that even near-optimal solutions are difficult to achieve. The inapproximability is proven using a reduction to the hitting set problem, a known NP-hard problem.

Main Results

The paper presents two main results:

1. Theoretical Intractability: The minimal controllability problem, the task of finding the sparsest vector or diagonal matrix to render a system controllable, is NP-hard. Moreover, it remains difficult to approximate within a multiplicative logarithmic factor. This is a significant theoretical result as it delineates the complexity boundaries of the problem.
2. Polynomial-time Greedy Approximation: On a more constructive note, the authors provide a polynomial-time approximation algorithm based on a greedy heuristic. This algorithm is capable of finding solutions that approximate the minimal controllability solution within a $c \log n$ multiplicative factor. The heuristic focuses on maximizing the rank increase of the controllability matrix upon each step of variable selection.

Implications and Future Directions

This research has several implications:

• Control of Large-Scale Systems: The findings are directly applicable to various fields where large, complex networks need control actions that are sparse, such as in biochemical networks, smart grids, and traffic systems. The results provide a clear direction for where computational efforts may be most effectively placed or avoided due to intractability.
• Algorithm Development: The authors' greedy approach provides a practical method for achieving near-minimal controllability, which holds promise for large-scale applications. Future developments could aim to refine these algorithms further or explore alternative heuristic methods that exploit specific problem structures for efficiency gains.
• Theoretical Advances: The results open avenues for further theoretical investigation into the properties of sparsity in networked systems and the potential for developing improved approximation algorithms under different conditions or assumptions of system structure.

Numerical Experiments

Numerical simulations with Erdos-Renyi random graphs are conducted to assess the practical efficacy of the proposed greedy algorithm for vector controllability. The results are promising, indicating that in many instances, a very sparse set of control inputs can achieve controllability. Notably, the greedy heuristic successfully finds a minimally sparse vector for a wide range of random graph instances, though the theoretical underpinning for the observed performance remains an open question.

Conclusion

This work represents a substantive contribution to the field of control theory, particularly in the context of scalability and complexity. By establishing the NP-hard nature of minimal controllability approximation and providing a practical, albeit approximate, solution method, it sets a foundation for both further theoretical exploration and practical applications in controlling large-scale dynamic networks. The findings could potentially inform those working with complex systems on the efficiencies and limitations of current methods, and pave the way for future research bridging combinatorial optimization and control theory.