The grounded Laplacian matrix $\LL_{-S}$ of a graph $\calG=(V,E)$ with $n=|V|$ nodes and $m=|E|$ edges is a $(n-s)\times (n-s)$ submatrix of its Laplacian matrix $\LL$, obtained from $\LL$ by deleting rows and columns corresponding to $s=|S| \ll n $ ground nodes forming set $S\subset V$. The smallest eigenvalue of $\LL_{-S}$ plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding $k \ll n$ edges among all the nonexistent edges forming the candidate edge set $Q = (V\times V)\backslash E$, in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set $Q$ to be $(S\times (V\backslash S))\backslash E$, and prove that it has the same optimal solution as the original problem, although the size of set $Q$ is reduced from $O(n^2)$ to $O(n)$. Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio $(1-e^{-\alpha\gamma})/\alpha$ and time complexity $O(kn^4)$, where $\gamma$ and $\alpha$ are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time $\tilde{O}(km)$, where $\tilde{O}(\cdot)$ suppresses the ${\rm poly} (\log n)$ factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.

翻译：图 $\calG=(V,E)$ 的接地拉普拉斯矩阵 $\LL_{-S}$ 是其拉普拉斯矩阵 $\LL$ 的一个 $(n-s)\times (n-s)$ 子矩阵，其中 $n=|V|$ 为节点数，$m=|E|$ 为边数。该子矩阵通过删除 $\LL$ 中对应于 $s=|S| \ll n$ 个接地节点（构成集合 $S\subset V$）的行和列得到。$\LL_{-S}$ 的最小特征值在实际场景中具有重要作用，例如表征领导者-跟随者意见动态的收敛速度，特征值越大表示意见收敛越快。本文研究从所有不存在的边构成的候选边集 $Q = (V\times V)\backslash E$ 中添加 $k \ll n$ 条边，以最大化接地拉普拉斯矩阵最小特征值的问题。我们证明该组合优化问题的目标函数是单调但非子模的。为求解该问题，我们首先将候选边集 $Q$ 限制为 $(S\times (V\backslash S))\backslash E$ 以简化问题，并证明该简化问题与原问题具有相同的最优解，尽管 $Q$ 的规模从 $O(n^2)$ 降至 $O(n)$。随后，我们提出两种贪心近似算法：其一是简单贪心算法，具有近似比 $(1-e^{-\alpha\gamma})/\alpha$ 和时间复杂度 $O(kn^4)$，其中 $\gamma$ 和 $\alpha$ 分别为子模比和曲率（给出了某些特殊情形下的界）；其二是无近似保证的快速贪心算法，运行时间为 $\tilde{O}(km)$，其中 $\tilde{O}(\cdot)$ 忽略 ${\rm poly} (\log n)$ 因子。我们在多种真实网络上进行大量实验，验证了所提算法在有效性和效率上的优越性。