Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

We provide a list of new natural VNP-intermediate polynomial families, based on basic (combinatorial) NP-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in VNP, and under the plausible hypothesis Mod p P ⫅̸ P/poly, are neither VNP-hard (even...

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Published in:Theory of computing systems Vol. 62; no. 3; pp. 622 - 652
Main Authors: Mahajan, Meena, Saurabh, Nitin
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2018
Springer Nature B.V
Subjects:
Combinatorial analysis
Complexity theory
Computer Science
Homomorphisms
Lower bounds
Polynomials
Theory of Computation
Homomorphisms
Lower bounds
Extension complexity
VBP
Completeness
VP
Tree decomposition
VNP-intermediate
Monotone projections
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary:We provide a list of new natural VNP-intermediate polynomial families, based on basic (combinatorial) NP-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in VNP, and under the plausible hypothesis Mod p P ⫅̸ P/poly, are neither VNP-hard (even under oracle-circuit reductions) nor in VP. Prior to this, only the Cut Enumerator polynomial was known to be VNP-intermediate, as shown by Bürgisser in 2000. We show next that over rationals and reals, the clique polynomial cannot be obtained as a monotone p -projection of the permanent polynomial, thus ruling out the possibility of transferring monotone clique lower bounds to the permanent. We also show that two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. These results augment recent results along this line due to Grochow. Finally, we describe a (somewhat natural) polynomial defined independent of a computation model, and show that it is VP-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established VP-completeness of a related polynomial but under constant-depth oracle circuit reductions. Both polynomials are based on graph homomorphisms. A simple restriction yields a family similarly complete for VBP.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-016-9740-y