On the Complexity and Approximability of Optimal Sensor Selection for Kalman Filtering

TL;DR

This paper proves that selecting optimal sensors for Kalman filtering is NP-hard and cannot be approximated within any constant factor, even for simple systems, highlighting the problem's computational difficulty.

Contribution

The paper establishes NP-hardness and non-approximability results for sensor selection in Kalman filtering, even in stable systems with identical sensor costs.

Findings

Sensor selection problem is NP-hard even for stable, identical-cost systems.

No constant-factor polynomial-time approximation exists for this problem.

Greedy algorithms can perform arbitrarily poorly in this context.

Abstract

Given a linear dynamical system, we consider the problem of selecting (at design-time) an optimal set of sensors (subject to certain budget constraints) to minimize the trace of the steady state error covariance matrix of the Kalman filter. Previous work has shown that this problem is NP-hard for certain classes of systems and sensor costs; in this paper, we show that the problem remains NP-hard even for the special case where the system is stable and all sensor costs are identical. Furthermore, we show the stronger result that there is no constant-factor (polynomial-time) approximation algorithm for this problem. This contrasts with other classes of sensor selection problems studied in the literature, which typically pursue constant-factor approximations by leveraging greedy algorithms and submodularity of the cost function. Here, we provide a specific example showing that greedy algorithms can perform arbitrarily poorly for the problem of design-time sensor selection for Kalman filtering.