Breaking O(nr) for Matroid Intersection

We present algorithms that break the $\tilde{O}(nr)$-independence-query bound for the Matroid Intersection problem for the full range of $r$; where $n$ is the size of the ground set and $r\le n$ is the size of the largest common independent set. The $\tilde{O}(nr)$ bound was due to the efficient implementations [CLSSW FOCS’19; Nguyen 2019] of the classic algorithm of Cunningham [SICOMP’86]. It was recently broken for large $r$ ($r=\omega (\sqrt{n})$), first by the $\tilde{O}(n^{1.5}\mathrm{/}{ϵ}^{1.5})$-query $(1-ϵ)$-approximation algorithm of CLSSW [FOCS’19], and subsequently by the $\tilde{O}(n^{6\mathrm{/}5}{r}^{3\mathrm{/}5})$-query exact algorithm of BvdBMN [STOC’21]. No algorithm, even an approximation one, was known to break the $\tilde{O}(nr)$ bound for the full range of $r$. We present an $\tilde O(n\sqrt{r}/\epsilon)$-query$(1-\epsilon)$-approximationalgorithmandan$\tilde O(nr^{3/4})$-queryexactalgorithm\mathrm{.}Ouralgorithmsimprovethe$\tilde O(nr)$boundandalsotheboundsbyCLSSWandBvdBMNforthefullrangeof$r$.