On the Computational Complexity of the Secure State-Reconstruction Problem

Abstract

In this paper, we discuss the computational complexity of reconstructing the state of a linear system from sensor measurements that have been corrupted by an adversary. The first result establishes that the problem is, in general, NP-hard. We then introduce the notion of eigenvalue observability and show that the state can be reconstructed in polynomial time when each eigenvalue is observable by at least 2s + 1 sensors and at most s sensors are corrupted by an adversary. However, there is a gap between eigenvalue observability and the possibility of reconstructing the state despite attacksthis gap has been characterized in the literature by the notion of sparse observability. To better understand this, we show that when the A matrix of the linear system has unitary geometric multiplicity, the gap disappears, i.e., eigenvalue observability coincides with sparse observability, and there exists a polynomial time algorithm to reconstruct the state provided the state can be reconstructed.

Decentralized Secure State-Tracking in Multi-Agent Systems

“…Although it has been known for long that the SSR problem is NP-hard [12], much progress was reported on reducing the computational complexity of solving the SSR problem. Many works, such as [10,[12][13][14], carve out subsets of the SSR problem instances which allow for a polynomial-time solution. More results on reducing the computational complexity of the SSR problem can be found in [15] and [16].…”