scholarly journals The ideal membership problem and polynomial identity testing

2010 ◽  
Vol 208 (4) ◽  
pp. 351-363 ◽  
Author(s):  
V. Arvind ◽  
Partha Mukhopadhyay
Keyword(s):  
Polynomial Identity ◽  
Membership Problem ◽  
Polynomial Identity Testing ◽  
Identity Testing ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

scholarly journals Gröbner–Shirshov Bases for Replicated Algebras

2017 ◽  
Vol 24 (04) ◽  
pp. 563-576 ◽  
Author(s):  
P.S. Kolesnikov
Keyword(s):  
Lie Algebras ◽  
Word Problem ◽  
Membership Problem ◽  
Leibniz Algebras ◽  
Universal Approach ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

scholarly journals The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

2021 ◽  
Vol 17 (4) ◽  
pp. 1-29
Author(s):  
Monaldo Mastrolilli
Keyword(s):  
Sufficient Conditions ◽  
Constraint Satisfaction Problems ◽  
Algorithmic Problem ◽  
Membership Problem ◽  
Efficient Computation ◽  
Constrained Problems ◽  
Constraint Language ◽  
Ideal Membership ◽  
Ideal Membership Problem ◽  
The Ideal

Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if f ϵ I . This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the IMP for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer’s dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems (CSPs) over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body Semi-Definite Programming (SDP) relaxations, identifying therefore the borderline of tractability for constraint language problems. This article is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares (SoS) proofs [O’Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the IMP tractability for combinatorial ideals implies bounded coefficients in SoS proofs.

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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