For a connected graph \(\mathcal {G}=(V,E)\) with n nodes, m edges, and Laplacian matrix \({\varvec{ L }}\) , a grounded Laplacian matrix \({\varvec{ L }}(S)\) of \(\mathcal {G}\) is a \((n-k) \times (n-k)\) principal submatrix of \({\varvec{ L }}\) , obtained from \({\varvec{ L }}\) by deleting k rows and columns corresponding to k selected nodes forming a set \(S\subseteq V\) . The smallest eigenvalue \(\lambda (S)\) of \({\varvec{ L }}(S)\) plays a pivotal role in various dynamics defined on \(\mathcal {G}\) . For example, \(\lambda (S)\) characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger \(\lambda (S)\) corresponding to smaller convergence time or better effectiveness of a pinning scheme. In this paper, we focus on the problem of optimally selecting a subset S of fixed \(k \ll n\) nodes, in order to maximize the smallest eigenvalue \(\lambda (S)\) of the grounded Laplacian matrix \({\varvec{ L }}(S)\) . We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty of obtaining the optimal solution, we first propose a naïve heuristic algorithm selecting one optimal node at each time for k iterations. Then we propose a fast heuristic scalable algorithm to solve this problem, using the derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our naïve heuristic algorithm takes \(\tilde{O}(knm)\) time, while the fast greedy heuristic has a nearly linear time complexity of \(\tilde{O}(km)\) , where \(\tilde{O}(\cdot )\) notation suppresses the \(\textrm{poly} (\log n)\) factors. We also conduct numerous experiments on different networks sized up to one million nodes, demonstrating the superiority of our algorithm in terms of efficiency and effectiveness compared to baseline methods.
