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...
Uložené v:
| Vydané v: | Computational complexity Ročník 29; číslo 2 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Cham
Springer International Publishing
01.12.2020
|
| Predmet: | |
| ISSN: | 1016-3328, 1420-8954 |
| On-line prístup: | Získať plný text |
| Tagy: |
Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
|
| Shrnutí: | 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 |