Sensitivity Oracles for Matroid Packing, Matroid Covering, and Matching Problems with Applications
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 $\alpha$-factor, with perfect matching as the case $\alpha=1$. We further introduce the \emph{subset sensitivity model}, in which updates are confined to a susceptible edge set of size $\sigma$ 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 $\widetilde O(f^\omega)$ query time and $O(f\sigma^2)$ space. We also prove a matching $\Omega(\min\{\sigma^2,n^2\})$-bit lower bound when $f\ge2$, establishing optimality.