Determinants vs. Algebraic Branching Programs

Article Type

Research Article

Publication Title

Computational Complexity

Abstract

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 (Chatterjee et al. 2022; Kumar2019), and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor (Kumar& Volk 2022). While determinantal complexity and ABP complexity are classically known to be polynomially equivalent (Mahajan & 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.

DOI

10.1007/s00037-024-00258-z

Publication Date

12-1-2024

Comments

Open Access; Green Open Access

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