Abstract
We present polynomial families complete for the well-studied algebraic complexity classes \(\textsf {VF}\), \(\textsf {VBP}\), \(\textsf {VP}\), and \(\textsf {VNP}\). The polynomial families are based on the homomorphism polynomials studied in the recent works of Durand et al. (2014) and Mahajan et al. [10]. We consider three different variants of graph homomorphisms, namely injective homomorphisms, directed homomorphisms and injective directed homomorphisms and obtain polynomial families complete for \(\textsf {VF}\), \(\textsf {VBP}\), \(\textsf {VP}\), and \(\textsf {VNP}\) under each one of these. The polynomial families have the following properties:
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The polynomial families complete for \(\textsf {VF}\), \(\textsf {VBP}\), and \(\textsf {VP}\) are model independent, i.e. they do not use a particular instance of a formula, ABP or circuit for characterising \(\textsf {VF}\), \(\textsf {VBP}\), or \(\textsf {VP}\), respectively.
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All the polynomial families are hard under p-projections.
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- 1.
The hardness is shown with respect to p-projection reductions. We will define them formally in Sect. 2.
- 2.
Valiant [1] raised the question of whether the Permanent is computable in \(\textsf {VP}\). This question is equivalent to asking whether \(\textsf {VP}\) \(=\) \(\textsf {VNP}\), which is the algebraic analogue of the P vs. NP question.
- 3.
Note that if we set all Y variables to 1, then this polynomial essentially counts the number of homomorphisms from G to H.
- 4.
We do not use c-reductions in this work. They are more general than p-reductions. The formal definition can be found in [6].
- 5.
It is a layer preserving isomorphic copy which maps the root node of \(\mathsf{MAT}_{{i}}\) to the root of \(H_n\).
- 6.
Recall that \(m(n) = 2c \lceil \log n\rceil + 1\), which is odd. Also this is without loss of generality.
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Chaugule, P., Limaye, N., Varre, A. (2019). Variants of Homomorphism Polynomials Complete for Algebraic Complexity Classes. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_8
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