Breaking O(nr) for Matroid Intersection
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We present algorithms that break the Õ(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 Õ(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 Õ(n^1.5/ϵ^1.5)-query (1-ϵ)-approximation algorithm of CLSSW [FOCS'19], and subsequently by the Õ(n^6/5r^3/5)-query exact algorithm of BvdBMN [STOC'21]. No algorithm, even an approximation one, was known to break the Õ(nr) bound for the full range of r. We present an Õ(n√(r)/ϵ)-query (1-ϵ)-approximation algorithm and an Õ(nr^3/4)-query exact algorithm. Our algorithms improve the Õ(nr) bound and also the bounds by CLSSW and BvdBMN for the full range of r.
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