AI 中文总结
该研究提出一种基于有效集的简单算法，将ACL ε-近似PageRank向量的计算时间复杂度优化为Õ(1/ε²)，并应用于ℓ₁-正则化PageRank优化，改进了现有边界的依赖关系。
AI 中文摘要
局部图聚类旨在在不探索整个图的情况下，在给定种子节点附近找到一个连通性良好的簇。Andersen、Chung和Lang（ACL；Internet Math. 2007）的经典局部聚类算法的关键步骤是从种子节点近似PageRank向量。他们的局部推送方法以 teleportation 参数 α 计算 ACL ε-近似PageRank向量，时间复杂度为 O(1/(αε))。我们提出一种算法，以高概率在 Õ(1/ε²) 时间内计算ACL ε-近似PageRank向量，该时间复杂度与图规模无关，仅对1/α存在多对数依赖，尽管对1/ε存在二次依赖。作为直接结果，我们获得了局部图聚类中目标 conductance 和目标 volume 之间的新运行时间权衡。我们的方法也适用于 ℓ₁-正则化PageRank的优化问题，以多对数依赖于1/α的方式计算加性近似极小值，改进了Martínez-Rubio、Wirth和Pokutta（COLT 2023）先前边界中1/√α的依赖。我们的算法基于一种直观的过程，该过程维护一个不断增长的节点有效集：它对当前集执行推送操作直至收敛，必要时扩展该集并重复此过程。我们证明，对于每个有效集，对应的极限状态是该集上对称对角占优（SDD）线性系统的解。我们将近似线性时间的SDD求解器应用于这些系统，并证明该近似保留了推送过程的特性。
英文摘要
Local graph clustering aims to find a well-connected cluster near a given seed node without exploring the entire graph. A key step in the classic local clustering algorithm of Andersen, Chung, and Lang (ACL; Internet Math. 2007) is to approximate the PageRank vector from the seed node. Their local push method computes an ACL $\varepsilon$-approximate PageRank vector with teleportation parameter $α$ in $O\bigl(1/(α\varepsilon)\bigr)$ time. We give an algorithm that computes an ACL $\varepsilon$-approximate PageRank vector in $\widetilde{O}\bigl(1 / \varepsilon^2\bigr)$ time with high probability. This bound is independent of the graph size and has only a polylogarithmic dependence on $1 / α$, albeit with a quadratic dependence on $1 / \varepsilon$. As a direct consequence, we obtain a new running-time tradeoff between the target conductance and target volume in local graph clustering. Our method also applies to the optimization problem of $\ell_1$-regularized PageRank and computes an additive approximate minimizer with a polylogarithmic dependence on $1/α$, improving the $1/\sqrtα$ dependence in the previous bound of Martínez-Rubio, Wirth, and Pokutta (COLT 2023). Our algorithm is based on an intuitive process that maintains a growing active set of nodes: it performs push operations on the current set until convergence and then expands the set and repeats the process if necessary. We show that for each active set, the corresponding limiting state is the solution to a symmetric diagonally dominant (SDD) linear system on the set. We apply nearly-linear-time SDD solvers to these systems and prove that the approximation preserves the properties of the push process.
Comments17 pages
