Towards a Converse for the Nearest Lattice Point Problem

TL;DR

This paper establishes a tight lower bound on the communication complexity for interactively computing the minimum of two variables uniformly distributed on a unit square, advancing understanding of lattice point problems.

Contribution

It derives a tight lower bound on the communication complexity of a key step in finding the nearest lattice point, using novel partition constraints and self-similarity analysis.

Findings

Lower bound on communication complexity is tight.

Characterization of partition constraints for optimal solutions.

Self-similarity property of the optimal interactive code.

Abstract

Upper bounds on the communication complexity of finding the nearest lattice point in a given lattice $\Lambda \subset \mathbb{R}^2$ was considered in earlier works~\cite{VB:2017}, for a two party, interactive communication model. Here we derive a lower bound on the communication complexity of a key step in that procedure. Specifically, the problem considered is that of interactively finding $\min(X_1,X_2)$, when $(X_1,X_2)$ is uniformly distributed on the unit square. A lower bound is derived on the single-shot interactive communication complexity and shown to be tight. This is accomplished by characterizing the constraints placed on the partition generated by an interactive code and exploiting a self similarity property of an optimal solution.