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≤ 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=ω(√n)), first by the tilde O(n1.5/ε1.5)-query (1-ε)-approximation algorithm of CLSSW [FOCS'19], and subsequently by the tilde O(n6/5r3/5)-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√r/ε)-query (1-ε)-approximation algorithm and an tilde O(nr3/4)-query exact algorithm. Our algorithms improve the tilde O(nr) bound and also the bounds by CLSSW and BvdBMN for the full range of r.