Abstract
The determinant is a canonical \(\textsf {VBP}\)-complete polynomial in the algebraic complexity setting. In this work, we introduce two variants of the determinant polynomial which we call \(\mathtt{StackDet}_n(X)\) and \(\mathtt{CountDet}_n(X)\) and show that they are \(\textsf {VP}\) and \(\textsf {VNP}\) complete respectively under p-projections. The definitions of the polynomials are inspired by a combinatorial characterisation of the determinant developed by Mahajan and Vinay (SODA 1997). We extend the combinatorial object in their work, namely clow sequences, by introducing additional edge labels on the edges of the underlying graph. The idea of using edge labels is inspired by the work of Mengel (MFCS 2013).
N. Limaye—Funded by SERB Project no. MTR/2017/000909.
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Notes
- 1.
A problem P is said to be complete for a Boolean complexity class \(\mathcal {C}\) if \(P \in \mathcal {C}\) and any problem \(P'\) in \(\mathcal {C}\) reduces to P in polynomial time.
- 2.
A polynomial \(P_n(X)\) is said to be complete for an algebraic complexity class \(\mathcal {A}\) if \(P_n(X)\) can be computed in \(\mathcal {A}\) and any polynomial \(P'_m(Y)\) can be obtained from \(P_n(X)\) by setting the variables in X to variables in Y or field constants. For formal definitions see Sect. 2.
- 3.
See also [3] for interesting variants of homomorphism polynomials.
- 4.
An algebraic branching program (ABP) is a directed layered acyclic graph with a source s and a sink t. The edges are labelled with formal variables or field constants. The weight of an s to t path \(\pi \) is the product of the weights on the edges of \(\pi \). The polynomial computed by the ABP is the sum of weights of all the s to t paths. A family \(f_n\) with s(n) number of variables and degree d(n) where both s(n) and d(n) are polynomially bounded in n is said to be in \(\textsf {VBP}\) iff there exist algebraic branching program of size polynomially bounded in n which computes \(f_n\). For more details see [11].
- 5.
\(\mathtt{No{\text {-}}op}\) stands for No-operation.
- 6.
A multiplication gate \(\alpha \) with children gates \(\alpha _{\ell }\) and \(\alpha _{r}\) in an arithmetic circuit C is called multiplicatively disjoint if the subcircuits rooted at \(\alpha _{\ell }\) and \(\alpha _{r}\) are disjoint.
- 7.
The depth of the circuit is the length of the longest input gate to output gate path.
- 8.
- 9.
As \(\varDelta \) is a power of 2, it is easy to note that adding three new vertices will always make the total number of vertices of graph \(G_N\) a multiple of 4.
- 10.
It is not too hard to see that \(\varDelta +4\) is a multiple of 4. (As \(\varDelta \) is a power of 2.).
- 11.
Note that the \(\textsf {VNP}\)-hardness can also possibly be shown in a single step via constructing a \(\textsf {RABP}\) \(\mathcal {R}_n\) which computes a polynomial family \(\mathcal {P}_n\) (known to be hard for \(\textsf {VNP}\) for all fields) and converting it to a counter graph \(G_n\) such that \(\mathtt{CountDet}_n\) defined over \(G_n\) computes \(\mathcal {P}_n\). However, constructing an \(\textsf {RABP}\) \(\mathcal {R}_n\) (and converting it then to a counter graph \(G_n\)) such that the function \(\phi \) in \(G_n\) getting exactly mapped to the function \(\varPhi \) defined in Definition 10 is not immediate. Therefore, we show the hardness in two steps.
- 12.
Recall that \(\mathtt{Perm}_m(Y) = \sum _{{\sigma : } \text { permutation of [m]}} \prod _{i \in [m]} y_{i,\sigma (i)}\).
- 13.
For an odd m, note that \(2m+2\) is always a multiple of 4.
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Chaugule, P., Limaye, N., Pandey, S. (2021). Variants of the Determinant Polynomial and the \(\textsf {VP}\)-Completeness. In: Santhanam, R., Musatov, D. (eds) Computer Science – Theory and Applications. CSR 2021. Lecture Notes in Computer Science(), vol 12730. Springer, Cham. https://doi.org/10.1007/978-3-030-79416-3_3
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