Maintaining a $k$-core decomposition quickly in a dynamic graph has important applications in network analysis. The main challenge for designing efficient exact algorithms is that a single update to the graph can cause significant global changes. Our paper focuses on \emph{approximation} algorithms with small approximation factors that are much more efficient than what exact algorithms can obtain. We present the first parallel, batch-dynamic algorithm for approximate $k$-core decomposition that is efficient in both theory and practice. Our algorithm is based on our novel parallel level data structure, inspired by the sequential level data structures of Bhattacharya et al [STOC '15] and Henzinger et al [2020]. Given a graph with $n$ vertices and a batch of updates $\mathcal{B}$, our algorithm provably maintains a $(2 + \varepsilon)$-approximation of the coreness values of all vertices (for any constant $\varepsilon > 0$) in $O(|\mathcal{B}|\log^2 n)$ amortized work and $O(\log^2 n \log\log n)$ depth (parallel time) with high probability. As a by-product, our $k$-core decomposition algorithm also gives a batch-dynamic algorithm for maintaining an $O(\alpha)$ out-degree orientation, where $\alpha$ is the current arboricity of the graph. We demonstrate the usefulness of our low out-degree orientation algorithm by presenting a new framework to formally study batch-dynamic algorithms in bounded-arboricity graphs. Our framework obtains new provably-efficient parallel batch-dynamic algorithms for maximal matching, clique counting, and vertex coloring. We implemented and experimentally evaluated our $k$-core decomposition algorithm on a 30-core machine with two-way hyper-threading on $11$ graphs of varying densities and sizes. [...]

翻译：在动态图中快速维护$k$-核分解在网络分析中具有重要应用。设计高效精确算法的主要挑战在于，单次图更新可能导致全局性显著变化。本文聚焦于具有小近似因子的\emph{近似}算法，其效率远超精确算法所能达到的水平。我们提出了首个在理论和实践中均高效的并行批处理动态算法，用于近似$k$-核分解。该算法基于我们创新的并行层级数据结构，其灵感来源于Bhattacharya等人[STOC '15]和Henzinger等人[2020]的序列化层级数据结构。给定一个包含$n$个顶点的图以及批量更新集$\mathcal{B}$，我们的算法能够可证明地维护所有顶点核数的$(2 + \varepsilon)$-近似值（对任意常数$\varepsilon > 0$），且摊销工作量为$O(|\mathcal{B}|\log^2 n)$，深度（并行时间）为$O(\log^2 n \log\log n)$，并具有高概率保证。作为副产品，我们的$k$-核分解算法还提供了维护$O(\alpha)$出度定向的批处理动态算法，其中$\alpha$为当前图的树性度。我们通过提出一个新框架来形式化研究有界树性度图中的批处理动态算法，展示了低出度定向算法的实用性。该框架为最大匹配、团计数和顶点着色问题提供了新的可证明高效的并行批处理动态算法。我们在具有双向超线程的30核机器上，对11个不同密度和规模的图实现了并实验评估了我们的$k$-核分解算法。[...]