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- W4288018985 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 $SinI_{1}capI_{2}$ 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 $SinI_{i}$ in $indep$ 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 $rank$ time. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact $O(nrlog r indep)$ time algorithm. This improves upon the running time of $O(nr^{1.5} indep)$ due to Cunningham in 1986 and $tilde{O}(n^{2} indep+n^{3})$ due to Lee, Sidford, and Wong in 2015. We also provide two algorithms which compute a $(1-epsilon)$-approximate solution to matroid intersection running in times $tilde{O}(n^{1.5}/eps^{1.5} indep)$ and $tilde{O}((n^{2}r^{-1}epsilon^{-2}+r^{1.5}epsilon^{-4.5}) indep)$, respectively. These results improve upon the $O(nr/eps indep)$-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 $O(nsqrt{r}log n rank)$-time exact algorithm and an $O(nepsilon^{-1}log n rank)$-time algorithm which obtains a $(1-eps)$-approximation to the matroid intersection problem. The former result improves over the $tilde{O}(nr rankt+n^{3})$-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." @default.
- W4288018985 created "2022-07-26" @default.
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- W4288018985 date "2019-11-25" @default.
- W4288018985 modified "2023-09-29" @default.
- W4288018985 title "Faster Matroid Intersection" @default.
- W4288018985 doi "https://doi.org/10.48550/arxiv.1911.10765" @default.
- W4288018985 hasPublicationYear "2019" @default.
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