Exponential lower bound via exponential sums

2026-01-01

arXiv.org

AI Summary

This work investigates whether summing exponential-size algebraic circuits necessarily yields circuits of exponential size, thereby connecting the VP vs. VNP conjecture with the Shub–Smale τ-conjecture. By introducing tools from parameterized complexity theory, the paper establishes—under the assumption that the τ-conjecture holds—the first explicit exponential fixed-parameter tractable (fpt) lower bounds in the algebraic setting. It further demonstrates a connection between the parameterized algebraic complexity class VWₙᵦ⁰[P] and the collapse of the counting hierarchy. Additionally, the study characterizes the completeness of certain polynomial families, defined via specific ring coverings, for the class VW[F], offering a novel framework for proving lower bounds and delivering structural insights into algebraic complexity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Applications of cryptography

Abstract

Valiant's famous VP vs. VNP conjecture states that the symbolic permanent polynomial does not have polynomial-size algebraic circuits. However, the best upper bound on the size of the circuits computing the permanent is exponential. Informally, VNP is an exponential sum of VP-circuits. In this paper we study whether, in general, exponential sums (of algebraic circuits) require exponential-size algebraic circuits. We show that the famous Shub-Smale $\tau$-conjecture indeed implies such an exponential lower bound for an exponential sum. Our main tools come from parameterized complexity. Along the way, we also prove an exponential fpt (fixed-parameter tractable) lower bound for the parameterized algebraic complexity class VW$_{nb}^0$[P], assuming the same conjecture. VW$_{nb}^0$[P] can be thought of as the weighted sums of (unbounded-degree) circuits, where only $\pm 1$ constants are cost-free. To the best of our knowledge, this is the first time the Shub-Smale $\tau$-conjecture has been applied to prove explicit exponential lower bounds. Furthermore, we prove that when this class is fpt, then a variant of the counting hierarchy, namely the linear counting hierarchy collapses. Moreover, if a certain type of parameterized exponential sums is fpt, then integers, as well as polynomials with coefficients being definable in the linear counting hierarchy have subpolynomial $\tau$-complexity. Finally, we characterize a related class VW[F], in terms of permanents, where we consider an exponential sum of algebraic formulas instead of circuits. We show that when we sum over cycle covers that have one long cycle and all other cycles have constant length, then the resulting family of polynomials is complete for VW[F] on certain types of graphs.

Problem

• exponential sums
• algebraic circuits
• VP vs. VNP
• Shub-Smale τ-conjecture
• parameterized complexity

Innovation

• exponential lower bound
• Shub-Smale τ-conjecture
• parameterized complexity
• algebraic circuits
• VW[F] completeness