Approximate Nonmyopic Sensor Selection via Submodularity and Partitioning

As sensors become more complex and prevalent, they present their own issues of cost effectiveness and timeliness. It becomes increasingly important to select sensor sets that provide the most information at the least cost and in the most timely and efficient manner. Two typical sensor selection problems appear in a wide range of applications. The first type involves selecting a sensor set that provides the maximum information gain within a budget limit. The other type involves selecting a sensor set that optimizes the tradeoff between information gain and cost. Unfortunately, both require extensive computations due to the exponential search space of sensor subsets. This paper proposes efficient sensor selection algorithms for solving both of these sensor selection problems. The relationships between the sensors and the hypotheses that the sensors aim to assess are modeled with Bayesian networks, and the information gain (benefit) of the sensors with respect to the hypotheses is evaluated by mutual information. We first prove that mutual information is a submodular function in a relaxed condition, which provides theoretical support for the proposed algorithms. For the budget-limit case, we introduce a greedy algorithm that has a constant factor of $(1 - 1/e)$ guarantee to the optimal performance. A partitioning procedure is proposed to improve the computational efficiency of the algorithms by efficiently computing mutual information as well as reducing the search space. For the optimal-tradeoff case, a submodular–supermodular procedure is exploited in the proposed algorithm to choose the sensor set that achieves the optimal tradeoff between the benefit and cost in a polynomial-time complexity.