Faster Matroid Intersection

Abstract

In this paper we consider the classic matroid intersection problem: given two matroids M_1=(V, I_1) and M_2=(V, I_2) defined over a common ground set V, compute a set S∊I_1 ∩ I_2 of largest possible cardinality, denoted by r. We consider this problem both in the setting where each M_i is accessed through an independence oracle, i.e. a routine which returns whether or not a set S ∊ I_i in T_ind time, and the setting where each M_i is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of S in M_i in T_rank time. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact O(nr log r⋅T_ind) time algorithm. This improves upon previous best known running times of O(nr^1.5⋅T_ind) due to Cunningham in 1986 and Õ(n^2⋅T_ind+n^3) due to Lee, Sidford, and Wong in 2015. We also provide two algorithms which compute a (1-ε)-approximate solution to matroid intersection running in times Õ(n^1.5/ε^1.5⋅T_ind) and Õ((n^2r^-1ε^-2+r^1.5ε^-4.5)⋅T_ind), respectively. These results improve upon the O(nr/ε⋅T_ind) -time algorithm of Cunningham (noted recently by Chekuri and Quanrud). Given a rank oracle, we provide algorithms with even better dependence on n and r. We provide an O(n√rlog n⋅T_rank) -time exact algorithm and an O(nε^-1 log n⋅T_rank) -time algorithm which obtains a (1-ε) -approximation to the matroid intersection problem. The former result improves over the Õ(nr⋅T_rank+n^3) -time algorithm by Lee, Sidford, and Wong. The rank oracle is of particular interest as the matroid intersection problem with this oracle is a special case (via Edmond's minimax characterization of matroid intersection) of the submodular function minimization (SFM) problem with an evaluation oracle, and understanding SFM query complexity is an outstanding open question.