Minimal Structural Perturbations for Network Controllability: Complexity Analysis

TL;DR

This paper proves that optimizing minimal link or node modifications for network controllability is NP-hard, highlighting the computational difficulty in designing robust controllable networks.

Contribution

It establishes the NP-hardness of minimal cost link and vertex modifications affecting network controllability, and provides approximation results for these problems.

Findings

Proves NP-hardness of minimal link addition for controllability

Shows NP-hardness of minimal link deletion affecting controllability

Provides approximation results for related network design problems

Abstract

Link (edge) addition/deletion or sensor/actuator failures are common structural perturbations for real network systems. This paper is related to the computation complexity of minimal (cost) link insertion, deletion and vertex deletion with respect to structural controllability of networks. Formally, given a structured system, we prove that: i) it is NP-hard to add the minimal cost of links (including links between state variables and from inputs to state variables) from a given set of links to make the system structurally controllable, even with identical link costs or a prescribed input topology; ii) it is NP-hard to determine the minimal cost of links whose deletion deteriorates structural controllability of the system, even with identical link costs or when the removable links are restricted in input links. It is also proven that determining the minimal cost of inputs whose deletion causes structural uncontrollability is NP-hard in the strong sense. The reductions in their proofs are technically independent. These results may serve an answer to the general hardness of optimally designing (modifying) a structurally controllable network topology and of measuring controllability robustness against link/actuator failures. Some fundamental approximation results for these related problems are also provided.