Faster Matroid Intersection
release_seloq5nt4jgdxibgeyuwo6y44y
by
Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and
Sam Chiu-wai Wong
2019
Abstract
In this paper we consider the classic matroid intersection problem: given two
matroids _1=(V,_1) and _2=(V,_2) defined over a common
ground set V, compute a set S∈_1∩_2 of largest possible
cardinality, denoted by r. We consider this problem both in the setting where
each _i is accessed through an independence oracle, i.e. a routine which
returns whether or not a set S∈_i in time, and the setting
where each _i is accessed through a rank oracle, i.e. a routine which
returns the size of the largest independent subset of S in _i in
time.
In each setting we provide faster exact and approximate algorithms. Given an
independence oracle, we provide an exact O(nrlog r ) time algorithm.
This improves upon the running time of O(nr^1.5) due to Cunningham
in 1986 and Õ(n^2+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/^1.5) and Õ((n^2r^-1ϵ^-2+r^1.5ϵ^-4.5)
), respectively. These results improve upon the O(nr/)-time
algorithm of Cunningham as 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√(r)log n )-time exact algorithm and an
O(nϵ^-1log n )-time algorithm which obtains a
(1-)-approximation to the matroid intersection problem. The former result
improves over the Õ(nr +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 of the submodular function
minimization problem with an evaluation oracle.
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