Tight Lower Bounds for Algebraic Communication and Applications

2026-09-15

AI Summary

本文研究了代数通信复杂性问题，通过提出一种新的下界框架来解决多项式评估和集合识别问题，并应用于证明某些算法的通信复杂性下界。

Abstract

Communication complexity studies how much information must be exchanged to solve a problem whose input is split among several parties. The classical setting deals with Boolean inputs split between two parties. We study an algebraic variant, where the inputs are vectors over a field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Alice and Bob have inputs $X\in \mathbb{F}^n$ and $Y\in \mathbb{F}^n$, respectively. We consider two kinds of tasks: the polynomial evaluation problem (compute the value of a polynomial $g\in \mathbb{F}[X,Y]$), and the set-recognition problem (decide whether (X,Y) is in $S$, for $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$). In both settings, Alice and Bob send evaluations of polynomials depending only on their own inputs. In the set-recognition problem, a referee receives the messages and may apply polynomial tests to the messages received so far; the outcomes of these tests determine acceptance or rejection. The protocols may be deterministic or probabilistic. We study: - Upper bounds and reductions: We give non-trivial upper bounds for a range of natural polynomial evaluation and set-recognition problems and prove reductions between different problems, which help organize the landscape of the model. - A lower bound framework and tight lower bounds: Our main technical contribution is a general framework for proving lower bounds for algebraic set-recognition problems. We prove several probabilistic lower bounds for natural problems, giving tight or near-tight characterizations of their algebraic communication. - Applications of the framework: Finally, we give two applications of our framework: proving lower bounds for a class of left-to-right algebraic algorithms (algebraic scanners) and a more general algebraic computational setting inspired by the BSS model.

Problem

• Communication Complexity
• Algebraic Communication
• Polynomial Evaluation
• Set-Recognition
• Lower Bounds

Innovation

• algebraic communication
• lower bound framework
• set-recognition problems
• polynomial evaluation