  Graph sampling set selection, where a subset of nodes are chosen to collect
samples to reconstruct a smooth graph signal, is a fundamental problem in graph
signal processing (GSP). Previous works employ an unbiased least-squares (LS)
signal reconstruction scheme and select samples via expensive extreme
eigenvector computation. Instead, we assume a biased graph Laplacian
regularization (GLR) based scheme that solves a system of linear equations for
reconstruction. We then choose samples to minimize the condition number of the
coefficient matrix---specifically, maximize the smallest eigenvalue
$\lambda_{\min}$. Circumventing explicit eigenvalue computation, we maximize
instead the lower bound of $\lambda_{\min}$, designated by the smallest
left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we
first convert the optimization to a dual problem, where we minimize the number
of samples needed to align all Gershgorin disc left-ends at a chosen
lower-bound target $T$. Algebraically, the dual problem amounts to optimizing
two disc operations: i) shifting of disc centers due to sampling, and ii)
scaling of disc radii due to a similarity transformation of the matrix. We
further reinterpret the dual as an intuitive disc coverage problem bearing
strong resemblance to the famous NP-hard set cover (SC) problem. The
reinterpretation enables us to derive a fast approximation scheme from a known
SC error-bounded approximation algorithm. We find an appropriate target $T$
efficiently via binary search. Extensive simulation experiments show that our
disc-based sampling algorithm runs substantially faster than existing sampling
schemes and outperforms other eigen-decomposition-free sampling schemes in
reconstruction error.
