  We study the problem of distributed state estimation in a network of sensing
units that can exchange their measurements but the rate of communication
between the units is constrained. The units collect noisy, possibly only
partial observations of the unknown state; they are assisted by a relay center
which can communicate at a higher rate and schedules the exchange of
measurements between the units. We consider the task of minimizing the total
mean-square estimation error of the network while promoting balance between the
individual units' performances. This problem is formulated as the maximization
of a monotone objective function subject to a cardinality constraint. By
leveraging the notion of weak submodularity, we develop an efficient greedy
algorithm for the proposed formulation and show that the greedy algorithm
achieves a constant factor approximation of the optimal objective. Our
extensive simulation studies illustrate the efficacy of the proposed
formulation and the greedy algorithm.
