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 polynomial-time 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.

中文翻译：

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卡尔曼滤波最优传感器选择和攻击的复杂性和逼近性

给定受随机噪声影响的线性动态系统，我们考虑选择一组最佳传感器（在设计时）以最小化卡尔曼滤波器的先验或后验误差协方差的稳态轨迹的问题，受制于某些选择预算约束。我们展示了这个问题没有多项式时间常数因子逼近算法的基本结果。这与文献中研究的其他类别的传感器选择问题形成对比，后者通常通过利用贪婪算法和成本函数的子模块性（或超模块性）来追求常数因子近似。在这里，我们提供了一个具体示例，说明贪婪算法对于卡尔曼滤波的设计时传感器选择问题可能表现不佳。然后，我们研究在预定义的攻击预算约束下攻击（即移除）一组安装的传感器的问题，以最大化卡尔曼滤波器的先验或后验误差协方差的稳态轨迹。我们再次证明，没有多项式时间常数因子逼近算法可以解决这个问题，并特别说明贪婪算法的性能可能很差。