eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-01-24
27:1
27:13
10.4230/LIPIcs.ITCS.2024.27
article
Determinants vs. Algebraic Branching Programs
Chatterjee, Abhranil
1
Kumar, Mrinal
2
Volk, Ben Lee
3
Indian Statistical Institute, Kolkata, India
Tata Institute of Fundamental Research, Mumbai, India
Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
We show that for every homogeneous polynomial of degree d, if it has determinantal complexity at most s, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most O(d⁵s). Moreover, we show that for most homogeneous polynomials, the width of the resulting homogeneous ABP is just s-1 and the size is at most O(ds).
Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree [Mrinal Kumar, 2019; Prerona Chatterjee et al., 2022], and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor [Mrinal Kumar and Ben Lee Volk, 2022].
While determinantal complexity and ABP complexity are classically known to be polynomially equivalent [Meena Mahajan and V. Vinay, 1997], the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol287-itcs2024/LIPIcs.ITCS.2024.27/LIPIcs.ITCS.2024.27.pdf
Determinant
Algebraic Branching Program
Lower Bounds
Singular Variety