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

Abstract

Given a linear dynamical system affected by stochastic noise, we consider the problem of selecting an optimal set of sensors (at design-time) to minimize the trace of the steady state a priori or a posteriori error covariance of the Kalman filter, subject to certain selection budget constraints. We show the fundamental result that there is no polynomialtime constant-factor 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 (or supermodularity) 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. We then study the problem of attacking (i.e., removing) a set of installed sensors, under predefined attack budget constraints, to maximize the trace of the steady state a priori or a posteriori error covariance of the Kalman filter. Again, we show that there is no polynomial-time constant-factor approximation algorithm for this problem, and show specifically that greedy algorithms can perform arbitrarily poorly.

Approximate Supermodularity of Kalman Filter Sensor Selection

“…Remark 3. In [16], it was shown that it is impossible to obtain a non-trivial, universal (system-independent) bound on α unless P = NP. To do so, they provide a reduction from the problem of exact cover by 3-sets (X3C) that relies on a dynamical system with noiseless outputs.…”

“…In general, solving this constrained output selection problem, often known as sensor selection, is NP-hard [10]- [16]. Still, there exist instances for which its solution can be approximated.…”

“…However, approximation guarantees for these problems are scarce. In particular, typical guarantees for greedy search do not apply since these functions are not supermodular, a diminishing returns property used to obtain these resultssee [13]- [16] for counter-examples in the context of control theory. Moreover, though convex relaxations of these problems have been proposed, they too lack near-optimality certificates [15], [21]- [26].…”

“…Moreover, though convex relaxations of these problems have been proposed, they too lack near-optimality certificates [15], [21]- [26]. In fact, system-independent guarantees cannot be provided for optimizing the state estimation MSE unless P = NP [16].…”

Network Theoretic Analysis of Maximum a Posteriori Detectors for Optimal Input Detection

“…In recent years, researchers have proposed a number of model-based and heuristic approaches for detecting and mitigating attacks against the actuators and the sensors in the network (see [2] and the references therein). Despite the success of these studies in revealing the performance and the limitations of attack detection mechanisms, several challenges remain, particularly in distinguishing malicious signals from ambient data, selecting optimal sensor locations to maximize the detection performance [3,4], and deriving simple graphical rubrics to readily evaluate and optimize network security [5,6].…”