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
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Abstract	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.
AbstractList	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 ModpP ⫅̸ 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. 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.
Author	Mahajan, Meena Saurabh, Nitin
Author_xml	– sequence: 1 givenname: Meena surname: Mahajan fullname: Mahajan, Meena organization: The Institute of Mathematical Sciences – sequence: 2 givenname: Nitin surname: Saurabh fullname: Saurabh, Nitin email: nitin@imsc.res.in organization: The Institute of Mathematical Sciences
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Cites_doi	10.4086/toc.2010.v006a007 10.7155/jgaa.00382 10.1016/S0747-7171(87)80063-9 10.1016/0095-8956(90)90132-J 10.1016/j.jco.2006.09.006 10.1145/322326.322341 10.1016/j.tcs.2004.08.008 10.1145/2716307 10.1016/0304-3975(83)90110-X 10.1137/0212043 10.1137/S0097539794266766 10.1002/1098-2418(200010/12)17:3/4<260::AID-RSA5>3.0.CO;2-W 10.4153/CJM-1965-045-4 10.1016/S0304-3975(99)00183-8 10.1561/0400000039 10.1016/S0304-3975(02)00017-8 10.1016/S0304-3975(99)00220-0 10.1007/BF01157687 10.1007/BFb0045375 10.1145/321864.321877 10.1007/BF02579196 10.1007/978-3-642-32589-2_29 10.1093/acprof:oso/9780198528173.001.0001 10.1145/2591796.2591834 10.1145/800135.804419 10.1007/978-3-662-04179-6 10.1007/978-3-319-21275-3 10.1007/3-540-08342-1_37 10.1007/978-3-642-22006-7_59 10.1145/800105.803397 10.1007/978-3-642-39206-1_6 10.1145/1206035.1206036
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Snippet	We provide a list of new natural VNP-intermediate polynomial families, based on basic (combinatorial) NP-complete problems that are complete under parsimonious... We provide a list of new natural VNP-intermediate polynomial families, based on basic (combinatorial) NP-complete problems that are complete under parsimonious...
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SubjectTerms	Combinatorial analysis Complexity theory Computer Science Homomorphisms Lower bounds Polynomials Theory of Computation
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Title	Some Complete and Intermediate Polynomials in Algebraic Complexity Theory
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