For a connected graph $\mathcal{G}=(V,E)$ with $n$ nodes, $m$ edges, and Laplacian matrix $\boldsymbol{{\mathit{L}}}$, a grounded Laplacian matrix $\boldsymbol{{\mathit{L}}}(S)$ of $\mathcal{G}$ is a $(n-k) \times (n-k)$ principal submatrix of $\boldsymbol{{\mathit{L}}}$, obtained from $\boldsymbol{{\mathit{L}}}$ by deleting $k$ rows and columns corresponding to $k$ selected nodes forming a set $S\subseteq V$. The smallest eigenvalue $\lambda(S)$ of $\boldsymbol{{\mathit{L}}}(S)$ 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 $\boldsymbol{{\mathit{L}}}(S)$. We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty to obtain the optimal solution, we first propose a na\"{\i}ve heuristic algorithm selecting one optimal node at each time for $k$ iterations. Then we propose a fast heuristic scalable algorithm to approximately solve this problem, using derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our na\"{\i}ve heuristic algorithm takes $\tilde{O}(knm)$ time, while the fast greedy heuristic has a nearly linear time complexity of $\tilde{O}(km)$. 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.

翻译：对于具有 $n$ 个节点、$m$ 条边及拉普拉斯矩阵 $\boldsymbol{{\mathit{L}}}$ 的连通图 $\mathcal{G}=(V,E)$，其接地拉普拉斯矩阵 $\boldsymbol{{\mathit{L}}}(S)$ 是通过从 $\boldsymbol{{\mathit{L}}}$ 中删除与 $k$ 个选定节点（构成集合 $S\subseteq V$）对应的 $k$ 行和 $k$ 列得到的 $(n-k) \times (n-k)$ 阶主子矩阵。矩阵 $\boldsymbol{{\mathit{L}}}(S)$ 的最小特征值 $\lambda(S)$ 在 $\mathcal{G}$ 上的多种动力学过程中起关键作用。例如，$\lambda(S)$ 刻画了领导者-跟随者一致性的收敛速度，以及牵制控制问题中牵制方案的有效性，且 $\lambda(S)$ 越大对应越短的收敛时间或更好的牵制方案效果。本文聚焦于优化选择固定数量 $k \ll n$ 的节点子集 $S$ 以最大化接地拉普拉斯矩阵 $\boldsymbol{{\mathit{L}}}(S)$ 的最小特征值 $\lambda(S)$ 的问题。我们证明该优化问题是 NP-hard，且目标函数非次模但具有单调性。由于难以获得最优解，我们首先提出一种朴素启发式算法，该算法在 $k$ 次迭代中每次选择一个最优节点。随后，我们利用导数矩阵、矩阵扰动和拉普拉斯求解器作为工具，提出一种快速可扩展的启发式算法来近似求解该问题。朴素启发式算法的时间复杂度为 $\tilde{O}(knm)$，而快速贪婪启发式算法具有接近线性的时间复杂度 $\tilde{O}(km)$。我们还在不同规模的网络上进行了大量实验（节点数高达百万），结果表明我们的算法在效率和有效性方面均具有优越性。