Breaking O(nr) for Matroid Intersection

Abstract

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; Nguy ễn 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 6/5 r 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.