Maximizing the smallest eigenvalue of grounded Laplacian matrix

For a connected graph G = (V, E) with n nodes, m edges, and Laplacian matrix LL, a grounded Laplacian matrix LL(S) of G is a (n-k)x(n-k) principal submatrix of LL, obtained from LL by deleting k rows and columns corresponding to k selected nodes forming a set S subset of V. The smallest eigenvalue lambda(S) of LL(S) plays a pivotal role in various dynamics defined on 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 << n nodes, in order to maximize the smallest eigenvalue lambda(S) of the grounded Laplacian matrix LL(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 naive 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 naive heuristic algorithm takes (O) over bar (knm) time, while the fast greedy heuristic has a nearly linear time complexity of (O) over bar (km), where (O) over bar(center dot) notation suppresses the 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.