Lattice Variant of the Sensitivity Conjecture

TL;DR

This paper investigates a lattice-based variant of the Sensitivity Conjecture, constructing optimal colorings and establishing bounds that could advance understanding of the relationship between sensitivity measures in Boolean functions.

Contribution

It introduces a lattice variant of the Sensitivity Conjecture, constructs optimal colorings, and provides the first non-constant lower bound on the lattice sensitivity s(C).

Findings

Constructed a coloring with d=O(s(C)^2) separation.

Proved the optimality of this separation for a large class.

Established a reverse reduction linking the lattice variant to the original conjecture.

Abstract

The Sensitivity Conjecture, posed in 1994, states that the fundamental measures known as the sensitivity and block sensitivity of a Boolean function f, s(f) and bs(f) respectively, are polynomially related. It is known that bs(f) is polynomially related to important measures in computer science including the decision-tree depth, polynomial degree, and parallel RAM computation time of f, but little is known how the sensitivity compares; the separation between s(f) and bs(f) is at least quadratic and at most exponential. We analyze a promising variant by Aaronson that implies the Sensitivity Conjecture, stating that for all two-colorings of the d-dimensional lattice $\mathbb{Z}^d$, d and the sensitivity s(C) are polynomially related, where s(C) is the maximum number of differently-colored neighbors of a point. We construct a coloring with the largest known separation between d and s(C), in which $d=O(s(C)^2)$, and demonstrate that it is optimal for a large class of colorings. We also give a reverse reduction from the Lattice Variant to the Sensitivity Conjecture, and using this prove the first non-constant lower bound on s(C). These results indicate that the Lattice Variant can help further the limited progress on the Sensitivity Conjecture.