For any undirected graph $G=(V,E)$ and a set $E_W$ of candidate edges with $E\cap E_W=\emptyset$, the $(k,\gamma)$-spectral augmentability problem is to find a set $F$ of $k$ edges from $E_W$ with appropriate weighting, such that the algebraic connectivity of the resulting graph $H=(V,E\cup F)$ is least $\gamma$. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph's conductance and the mixing time of random walks in a graph, maximising the resulting graph's algebraic connectivity by adding a small number of edges has been studied over the past 15 years. In this work we present an approximate and efficient algorithm for the $(k,\gamma)$-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the $(k,\gamma)$-spectral augmentability problem. (1) We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [GB06]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly. (2) We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least $\Omega(n^2mk)$ time [KMST10]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces.

中文翻译：

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增强图的代数连通性

对于任何无向图 $G=(V,E)$ 和具有 $E\cap E_W=\emptyset$ 的候选边集合 $E_W$，$(k,\gamma)$-谱增广问题是找到一个用适当的权重从 $E_W$ 设置 $k$ 边的 $F$，使得结果图 $H=(V,E\cup F)$ 的代数连通性最小。由于代数连通性与许多其他图参数（包括图的电导和图中随机游走的混合时间）之间存在紧密联系，因此在过去的 15 年中研究了通过添加少量边来最大化结果图的代数连通性年。在这项工作中，我们为 $(k,\gamma)$-谱可增性问题提出了一种近似且有效的算法，并且我们的算法在广泛的参数范围内以几乎线性的时间运行。我们的主要算法基于论文中开发的以下两种新技术，它们可能具有超出 $(k,\gamma)$-谱可增性问题的应用。(1) 我们提出了一种快速算法，用于解决 [GB06] 中代数连通性最大化问题的 SDP 的可行性版本。我们的算法基于用于解决 SDP 的经典原始对偶框架，而后者又使用乘法权重更新算法。我们提出了一种统一不同矩阵和向量变量的 SDP 约束的新方法，并相应地给出了一个很好的分离预言机。(2) 我们为子图稀疏问题提出了一种有效的算法，并且对于范围广泛的参数，我们的算法在几乎线性的时间内运行，这与之前最著名的算法至少运行在 $\Omega(n^2mk) $ 时间 [KMST10]。