Exponential lower bound via exponential sums

TL;DR

The paper investigates whether exponential sums over algebraic circuits necessarily require exponential-size circuits, tying this to the Shub-Smale tau-conjecture. It introduces a parameterized algebraic framework and shows that a subexponential upper bound for certain exponential sums would collapse the linear counting hierarchy, while the tau-conjecture implies exponential lower bounds for explicit exponential-sum families. It develops the VW-hierarchy and a transfer mechanism that connects conditional collapses in VW to unconditional separations in VP versus VNP, and it constructs restricted permanents that are VW[F]-hard on particular graph classes. Finally, it provides upper-bound results for restricted permanents on bounded-treewidth graphs, illustrating a nuanced landscape where algebraic, parameterized, and counting-hierarchy considerations interact to yield conditional hardness and tractable cases.

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 $τ$-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 $τ$-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 $τ$-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.

Figures (9)

• Figure 1: The input gadget and the two ways how to cover it (drawn blue).
• Figure 2: The multiplication gadget. Iff-couplings are drawn as dashed bidirected edges.
• Figure 3: The addition gadget. If the corresponding gate has fanin $t$, then there are $t$ nodes at the bottom.
• Figure 4: Lefthand side: The covering (drawn blue) if the top node is not externally covered. Righthand side: The covering if the top node is externally covered. One of the bottome nodes is covered by a $2$-cycle. This is the child in the corresponding parse tree.
• Figure 5: The iff-gadget. The edges $(x,y)$ and $(u,v)$ are the iff-coupled edges in the original graph.

Theorems & Definitions (73)

• Theorem 1.3: Informal version of \ref{['thm:formal-exp-lb']}
• Theorem 1.4: Informal version of \ref{['thm:collpase-chsubexp']} and \ref{['thm:tau']}
• Theorem 1.5: $\mathsf{VW}[\mathsf{F}]$-Completeness
• Definition 2.1: Algebraic FPT classes
• Definition 2.2: Fpt-projection
• Definition 2.3: Fpt-substitution
• Lemma 2.4
• Definition 3.1
• Definition 3.2
• Definition 3.3