Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization

2014 ◽  
Author(s):  
Swastik Kopparty ◽  
Shubhangi Saraf ◽  
Amir Shpilka
Keyword(s):  
Polynomial Identity ◽  
Polynomial Factorization ◽  
Multivariate Polynomial ◽  
Polynomial Identity Testing ◽  
Identity Testing

Equivalence of Polynomial Identity Testing and Polynomial Factorization

2015 ◽  
Vol 24 (2) ◽  
pp. 295-331 ◽  
Author(s):  
Swastik Kopparty ◽  
Shubhangi Saraf ◽  
Amir Shpilka
Keyword(s):  
Polynomial Identity ◽  
Polynomial Factorization ◽  
Polynomial Identity Testing ◽  
Identity Testing

Parallel algorithms for matroid intersection and matroid parity

2015 ◽  
Vol 07 (02) ◽  
pp. 1550019
Author(s):  
Jinyu Huang
Keyword(s):  
Parallel Algorithms ◽  
Polynomial Identity ◽  
Black Box ◽  
Matroid Intersection ◽  
Polynomial Identity Testing ◽  
Identity Testing ◽  
Common Base ◽  
Graphic Matroid ◽  
Matroid Parity ◽  
Nc Algorithms

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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