Linear Matroid Intersection is in Quasi-NC

Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which...

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Bibliographic Details
Published in:Computational complexity Vol. 29; no. 2
Main Authors: Gurjar, Rohit, Thierauf, Thomas
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.12.2020
Subjects:
Algorithm Analysis and Problem Complexity
Computational Mathematics and Numerical Analysis
Computer Science
68W20
05B35
90C57
Matroid Intersection
Polynomial Identity Testing
68W10
Isolation Lemma
Derandomization
ISSN:1016-3328, 1420-8954
Online Access:Get full text
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Summary:Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size n O ( log n ) and O (polylog( n )) depth. Moreover, the depth of the circuit can be reduced to O (log 2 n ) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix , when the given matrix is of the form A 0 + A 1 x 1 + ⋯ + A m x m , for an arbitrary matrix A 0 and rank-1 matrices A 1 , A 2 , ⋯ , A m . This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.
ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-020-00200-z