Efficient algorithms for graphic matroid intersection and parity

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).

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Efficient algorithms for a family of matroid intersection problems

Journal of Algorithms ◽

10.1016/0196-6774(84)90042-7 ◽

1984 ◽

Vol 5 (1) ◽

pp. 80-131 ◽

Cited By ~ 49

Author(s):

Harold N Gabow ◽

Robert E Tarjan

Keyword(s):

Efficient Algorithms ◽

Matroid Intersection

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Efficient algorithms for simple matroid intersection problems

10.1109/sfcs.1979.14 ◽

1979 ◽

Author(s):

Harold N. Gabow ◽

Robert E. Tarjan

Keyword(s):

Efficient Algorithms ◽

Matroid Intersection ◽

Simple Matroid

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Tidal Flow: A Fast and Teachable Maximum Flow Algorithm

OLYMPIADS IN INFORMATICS ◽

10.15388/ioi.2018.03 ◽

2018 ◽

Vol 12 ◽

pp. 25-41

Author(s):

Matthew C. FONTAINE

Keyword(s):

Computer Science ◽

Tidal Flow ◽

Maximum Flow ◽

Efficient Algorithms ◽

Science Students ◽

Flow Algorithm ◽

Maximum Flows

Among the most interesting problems in competitive programming involve maximum flows. However, efficient algorithms for solving these problems are often difficult for students to understand at an intuitive level. One reason for this difficulty may be a lack of suitable metaphors relating these algorithms to concepts that the students already understand. This paper introduces a novel maximum flow algorithm, Tidal Flow, that is designed to be intuitive to undergraduate andpre-university computer science students.

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