We develop a new algorithmic framework for designing approximation algorithms for cut-based optimization problems on capacitated undirected graphs that undergo edge insertions and deletions. Specifically, our framework dynamically maintains a variant of the hierarchical $j$-tree decomposition of [Madry FOCS'10], achieving a poly-logarithmic approximation factor to the graph's cut structure and supporting edge updates in $O(n^ε)$ amortized update time, for any arbitrarily small constant $ε\in (0,1)$. Consequently, we obtain new trade-offs between approximation and update/query time for fundamental cut-based optimization problems in the fully dynamic setting, including all-pairs minimum cuts, sparsest cut, multi-way cut, and multi-cut. For the last three problems, these trade-offs give the first fully-dynamic algorithms achieving poly-logarithmic approximation in sub-linear time per operation. The main technical ingredient behind our dynamic hierarchy is a dynamic cut-sparsifier algorithm that can handle vertex splits with low recourse. This is achieved by white-boxing the dynamic cut sparsifier construction of [Abraham et al. FOCS'16], based on forest packing, together with new structural insights about the maintenance of these forests under vertex splits. Given the versatility of cut sparsification in both the static and dynamic graph algorithms literature, we believe this construction may be of independent interest.

翻译：我们提出了一种新的算法框架，用于设计在经历边插入与删除的带容量无向图上基于割的优化问题的近似算法。具体而言，该框架动态维护了[Madry FOCS'10]中层次化 $j$-树分解的一个变体，能够以多对数近似比逼近图的割结构，并支持在 $O(n^ε)$ 的摊还更新时间内处理边更新，其中 $ε\in (0,1)$ 为任意小的常数。基于此，我们在完全动态场景下为若干基础的基于割的优化问题（包括全对最小割、稀疏割、多路割与多割）获得了近似比与更新/查询时间之间的新权衡。对于后三个问题，这些权衡首次实现了在每次操作亚线性时间内达到多对数近似比的完全动态算法。我们动态层次结构背后的核心技术要素是一种能够以低代价处理顶点分割的动态割稀疏化算法。这是通过白盒化[Abraham et al. FOCS'16]基于森林包装的动态割稀疏化构造，并结合关于在顶点分割下维护这些森林的新颖结构洞察实现的。鉴于割稀疏化技术在静态与动态图算法研究中的广泛适用性，我们相信该构造可能具有独立的学术价值。