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、$m$条边和拉普拉斯矩阵$\boldsymbol{{\mathit{L}}}$的连通图$\mathcal{G}=(V,E)$，其接地拉普拉斯矩阵$\boldsymbol{{\mathit{L}}}(S)$是$\boldsymbol{{\mathit{L}}}$的一个$(n-k) \times (n-k)$阶主子矩阵，通过删除对应于$k$个选定的节点集合$S\subseteq V$的$k$行和列得到。$\boldsymbol{{\mathit{L}}}(S)$的最小特征值$\lambda(S)$在$\mathcal{G}$上的各种动力学中起着关键作用。例如，$\lambda(S)$刻画了领导-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，且较大的$\lambda(S)$对应更短的收敛时间或更好的牵制方案效果。本文聚焦于在固定$k \ll n$的条件下最优选择子集$S$的问题，以最大化接地拉普拉斯矩阵$\boldsymbol{{\mathit{L}}}(S)$的最小特征值$\lambda(S)$。我们证明该优化问题是NP难的，且目标函数非子模但单调。鉴于最优解难以获得，我们首先提出一种朴素启发式算法，在$k$次迭代中每次选择一个最优节点。进而，利用导数矩阵、矩阵扰动和拉普拉斯求解器作为工具，我们提出一种快速可扩展的启发式算法来近似求解该问题。我们的朴素启发式算法耗时$\tilde{O}(knm)$，而快速贪婪启发式算法具有近线性的时间复杂度$\tilde{O}(km)$。我们在不同规模（高达百万节点）的网络上进行了大量实验，结果表明了本算法在效率和有效性上的优越性。