发表机构
IIT Delhi（印度理工学院德里分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出统一代数框架，得到支持任意f次更新的拟阵打包等问题的灵敏度预言机，解决开放问题并证明下界，应用于流、割、匹配等优化问题。
AI 中文摘要
灵敏度预言机会对图进行预处理，使得在发生任意f条边的插入和删除后，无需从头重新计算即可回答查询。对于结构优化问题，现有研究进展有限：对于流和割，所有已知的紧凑预言机仅能处理f≤2次故障；现有的s-割和全局最小割预言机仅适用于无向图；而对于匹配、树形图和生成树打包以及 Arboricity（边覆盖数），尚无适用于f>1的高效预言机。我们提出了一种基于稀疏线性拟阵的打包、覆盖和奇偶性问题的灵敏度预言机的统一代数框架，得到了支持所有这些问题任意数量f次更新的首个预言机（所有构造均为随机蒙特卡洛型）。具体而言，我们获得了高效的精确(s,t)-最大流/最小割预言机，以接近最优空间解决了Baswana、Bhanja和Pandey（ICALP'22）提出的开放问题；得到了全对k-有界流预言机，推广了Brand和Saranurak（FOCS'19，k=1的情况）的接近最优可达性预言机；得到了首个适用于任意f次更新的有向s-割和全局最小割预言机；得到了k-不交树形图、k-不交生成树、彩色生成树和边覆盖数的预言机；以及α-因子存在性的预言机，其中完美匹配为α=1的情况。我们进一步引入了子集灵敏度模型，其中更新被限制在预处理时固定的大小为σ的易感边集中。在此，我们将更新与拟阵表示解耦，完全消除了对k和拟阵密度的依赖：上述所有问题均以Õ(f^ω)的查询时间和O(fσ²)的空间得到支持。当f≥2时，我们还证明了匹配的Ω(min{σ²,n²})位下界，确立了最优性。
英文摘要
Sensitivity oracles preprocess a graph so that queries can be answered after any $f$ edge insertions and deletions, without recomputing from scratch. For structural optimization problems the known landscape is limited: for flows and cuts, all known compact oracles handle only $f\le2$ failures; existing oracles for $s$- and global min-cut apply only to undirected graphs; and for matchings, arborescence and spanning-tree packings, and arboricity, no efficient oracle is known for $f>1$. We present a unified algebraic framework based on sensitivity oracles for matroid packing, covering, and parity of sparse linear matroids, yielding the first oracles supporting an arbitrary number $f$ of updates across all of these problems (all constructions randomized Monte-Carlo). Concretely, we obtain efficient oracles for exact $(s,t)$-max-flow/min-cut, resolving an open problem of Baswana, Bhanja, and Pandey (ICALP'22) with near-optimal space; for all-pairs $k$-bounded flow, generalizing the near-optimal reachability oracle of Brand and Saranurak (FOCS'19, the case $k=1$); the first oracles for any $f$ for directed $s$- and global min-cut; oracles for $k$-disjoint arborescences, $k$-disjoint spanning trees, colorful spanning trees, and arboricity; and oracles for the existence of an $α$-factor, with perfect matching as the case $α=1$. We further introduce the \emph{subset sensitivity model}, in which updates are confined to a susceptible edge set of size $σ$ fixed during preprocessing. Here we decouple updates from the matroid representation and eliminate the dependence on $k$ and the matroid density altogether: all of the above are supported with $\widetilde O(f^ω)$ query time and $O(fσ^2)$ space. We also prove a matching $Ω(\min\{σ^2,n^2\})$-bit lower bound when $f\ge2$, establishing optimality.
Comments65 pages, 3 figures
