Faster Matroid Intersection
Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and, Sam Chiu-wai Wong

TL;DR
This paper introduces faster algorithms for the matroid intersection problem, improving both exact and approximate solutions under independence and rank oracle models, with significant efficiency gains over previous methods.
Contribution
It presents new faster algorithms for matroid intersection, including exact and approximate solutions, with improved time complexities for both independence and rank oracle settings.
Findings
Exact algorithm with $O(nr\,\log r\,\indep)$ time
Approximate algorithms with $\tilde{O}(n^{1.5}/\eps^{1.5}\,\indep)$ and $\tilde{O}((n^{2}r^{-1}\eps^{-2}+r^{1.5}\eps^{-4.5})\,\indep)$ times
Exact rank oracle algorithm with $O(n\sqrt{r}\log n\,\rank)$ time
Abstract
In this paper we consider the classic matroid intersection problem: given two matroids and defined over a common ground set , compute a set of largest possible cardinality, denoted by . We consider this problem both in the setting where each is accessed through an independence oracle, i.e. a routine which returns whether or not a set in time, and the setting where each is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of in in time. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact time algorithm. This improves upon the running time of due to Cunningham in 1986 and $\tilde{O}(n^{2}…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Computational Geometry and Mesh Generation
