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arXiv 2610.16637cs.DS
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基于预测的动态最大独立集

Dynamic Maximal Independent Set using Predictions

Indian Institute of Technology Roorkee（印度理工学院罗尔基分校）

Shahbaz Khan, Anshul Kumar

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AI总结：

该研究在预测模型下提出了适用于增量、减量及完全动态场景的确定性动态最大独立集算法，其更新时间依赖于预测误差，满足带预测算法的必要属性，还可利用通用误差修正框架放宽误差度量并保持时间界。

摘要：

我们在预测模型下，针对具有$n$个顶点、最多$m$条边的图，在边更新场景中提出了首个动态最大独立集（MIS）算法，该模型中可在真实边更新在线到达前，对预测的更新序列进行离线预处理。我们考虑增量（仅插入）、减量（仅删除）及完全动态（插入与删除兼具）三种模型。无预测时，确定性算法在减量场景下最坏情况更新时间为$O(n)$，在增量及完全动态场景下为$O(m)$；Bernstein等人[SODA26]在组合BMM猜想下证明，针对自适应敌手的组合增量算法需要$\u03a9(n^{1-o(1)})$的摊销更新时间。我们提出的确定性增量、减量及完全动态算法，其更新时间依赖于预测误差，该误差通过误预测更新数量$\u03b7_e$、MIS成员身份被误预测的顶点数量$\u03b7_v$及这些顶点的总度数$\u03b7_v^*$衡量。在$O(mn)$预处理下，我们的减量算法最坏情况更新时间为$O(\u03b7_v)$，增量算法为$O(\u03b7_e+\u03b7_v^*)$；完全动态算法在$u$次更新的$O(un)$预处理下，最坏情况更新时间为$O(\u03b7_e+\u03b7_v^*)$。我们还给出了空间高效变体：减量场景下空间为$O(m)$，增量场景下空间为$O(m\ud835\udcc1 m)$，代价是最坏情况更新时间增加对数级开销。我们的算法满足一致性、鲁棒性及平滑退化，这是带预测算法的必要属性；它们还可利用Khan等人[Arxiv26]的通用误差修正框架，将误差度量从首个差异点放宽为前缀匹配（针对部分动态场景）和集合编辑距离（针对完全动态场景），同时保持相同的时间界。

英文摘要：

We present the first dynamic maximal independent set (MIS) of a graph with $n$ vertices and up to $m$ edges under edge updates in the prediction model, where a predicted update sequence is available for offline preprocessing before the real updates arrive online. We consider the incremental (insertion-only), decremental (deletions-only), and fully dynamic (insertions and deletions) models. Without predictions, deterministic algorithms need worst-case update time of $O(n)$ for decremental and $O(m)$ for incremental and fully dynamic settings. Bernstein et al.~[SODA26] proved under the combinatorial BMM conjecture, combinatorial incremental algorithms against an adaptive adversary require $Ω(n^{1-o(1)})$ amortized update time. We present deterministic incremental, decremental, and fully dynamic algorithms whose update times depend on the prediction error. This error is measured by the number $η_e$ of mispredicted updates, and by the number $η_v$ and total degree $η_v^*$ of vertices whose MIS membership was mispredicted. With $O(mn)$ preprocessing, our decremental and incremental algorithms achieve worst-case update times $O(η_v)$ and $O(η_e+η_v^*)$, respectively. Our fully dynamic algorithm achieves $O(η_e+η_v^*)$ worst-case update time with $O(un)$ preprocessing for $u$ updates. We also give space-efficient variants using $O(m)$ space for the decremental and $O(m\sqrt{m})$ for the incremental setting, at the cost of a logarithmic overhead in worst-case update time. Our algorithms satisfy consistency, robustness, and smooth degradation, the required properties of algorithms with predictions. They can also utilize the general error correction framework of Khan et al.~[Arxiv26], relaxing the error measures from first point of difference to prefix matching (for partially dynamic) and set edit distance (for fully dynamic), maintaining the same bounds.

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