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- W4300719219 abstract "We provide a list of new natural $mathsf{VNP}$-intermediate polynomial families, based on basic (combinatorial) $mathsf{NP}$-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in $mathsf{VNP}$, and under the plausible hypothesis $mathsf{Mod}_pmathsf{P} notsubseteq mathsf{P/poly}$, are neither $mathsf{VNP}$-hard (even under oracle-circuit reductions) nor in $mathsf{VP}$. Prior to this, only the Cut Enumerator polynomial was known to be $mathsf{VNP}$-intermediate, as shown by B{u}rgisser in 2000. We next show that over rationals and reals, two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. This augments 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 $mathsf{VP}$-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established $mathsf{VP}$-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 $mathsf{VBP}$." @default.
- W4300719219 created "2022-10-04" @default.
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- W4300719219 date "2016-03-15" @default.
- W4300719219 modified "2023-09-23" @default.
- W4300719219 title "Some Complete and Intermediate Polynomials in Algebraic Complexity Theory" @default.
- W4300719219 doi "https://doi.org/10.48550/arxiv.1603.04606" @default.
- W4300719219 hasPublicationYear "2016" @default.
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