Exponential lower bound via exponential sums

TL;DR

This paper explores the complexity of exponential sums of algebraic circuits, showing that under the Shub-Smale τ-conjecture, such sums require exponential-size circuits, and connects these bounds to broader complexity class implications.

Contribution

It demonstrates the first explicit exponential lower bounds for exponential sums assuming the τ-conjecture and links these bounds to collapses in the counting hierarchy.

Findings

Exponential sums of algebraic circuits require exponential-size circuits under the τ-conjecture.

Assuming the conjecture, fpt algorithms imply collapses in the linear counting hierarchy.

Characterization of VW[F] class via permanents and exponential sums over specific cycle covers.

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.