AI 中文总结
研究动态最小支配集问题，针对树宽有界的动态图，提出基于原始对偶框架的算法，能以O(α·log(Cn))更新时间维护O(α)近似MDS，改进了近似保证并摆脱对Δ的依赖，常数树宽图族可达O(1)近似和O(log(Cn))更新时间。
AI 中文摘要
在动态最小支配集（MDS）问题中，目标是在一个具有顶点成本在[1/C,1]的n顶点图中，高效地维护一个近似MDS，同时进行边的插入和删除操作。在STACS'19 [HIPS19]中表明，在无加权图中可以以O(Δ·log n)的更新时间维护一个O(log n)近似MDS，其中Δ是整个更新序列中最大度的上限，并且在STOC'23 [SU23]中，这被扩展到加权图，并将近似保证提高到(1 + ε)ln Δ。对于任何非平凡图族，是否有可能在不依赖于Δ的情况下实现poly(log n)的更新时间？这个基本问题即使在森林中以及对于无加权实例也仍然未解决。图G的树宽α = α(G)是其并集为G的边不相交森林的最小数量，并且是稀疏性的标准度量。虽然在任何图中α都由Δ界定，但各种实际图族在α和Δ之间表现出显著差距。在这项工作中，我们表明对于在整个更新序列中树宽由α界定的动态图，可以以O(α·log(Cn))的更新时间维护一个O(α)近似MDS。这用α取代了先前更新界限中对Δ的依赖，同时也改进了有界树宽图的近似保证。特别是，对于任何具有常数树宽的图族，我们的算法给出了具有O(log(Cn))更新时间的O(1)近似。为了实现这一结果，我们的算法偏离了先前基于贪心的方法，而是依赖于原始对偶框架和特定于有界树宽图的新结构见解。
英文摘要
In the dynamic {\em minimum dominating set (MDS)} problem, the goal is to efficiently maintain an approximate MDS in an $n$-vertex graph with vertex costs in $[1/C,1]$ undergoing edge insertions and deletions. In STACS'19 [HIPS19] it was shown that an $O(\log n)$-approximate MDS can be maintained in {\em unweighted graphs} with $O(Δ\cdot \log n)$ update time, where $Δ$ is an upper bound on the maximum degree throughout the update sequence, and in STOC'23 [SU23] this was extended to weighted graphs and improves the approximation guarantee to $(1+ε)\ln Δ$. Is it possible to achieve $\mathrm{poly}(\log n)$ update time without any dependence on $Δ$, for any nontrivial graph family? This basic question has remained open even in {\bf forests} and even for {\bf unweighted instances}. The {\em arboricity} $α=α(G)$ of a graph $G$ is the minimum number of edge-disjoint forests whose union is $G$, and is a standard measure of sparsity. While $α$ is bounded by $Δ$ in any graph, various real-world graph families exhibit a significant gap between $α$ and $Δ$. In this work, we show that one can maintain an $O(α)$-approximate MDS with update time $O(α\cdot \log (Cn))$, for dynamic graphs whose {\em arboricity} is bounded by $α$ throughout the update sequence. This replaces the dependence on $Δ$ in prior update bounds with $α$, while also improving the approximation guarantee for bounded-arboricity graphs. In particular, for any graph family of constant arboricity, our algorithm gives an $O(1)$-approximation with $O(\log (Cn))$ update time. To achieve this result, our algorithm departs from prior {\em greedy-based} approaches, relying instead on the {\em primal-dual framework} and new structural insights specific to bounded arboricity graphs.
CommentsESA'26
