Query complexity of Boolean functions on slices

TL;DR

This paper investigates the deterministic query complexity of Boolean functions on slices of the hypercube, revealing near-linear lower bounds and explicit constructions, and relating complexity to combinatorial structures like Johnson graphs and Ramsey graphs.

Contribution

It provides new lower bounds and explicit functions for query complexity on slices, and introduces a method to transfer complexity measures from hypercube functions to slice functions.

Findings

Existence of functions on the balanced slice requiring nearly n queries.

Explicit functions based on Johnson graphs with high query complexity.

Connections between hard functions on the weight-2 slice and Ramsey graphs.

Abstract

We study the deterministic query complexity of Boolean functions on slices of the hypercube. The $k^{th}$ slice $\binom{[n]}{k}$ of the hypercube $\{0,1\}^n$ is the set of all $n$-bit strings with Hamming weight $k$. We show that there exists a function on the balanced slice $\binom{[n]}{n/2}$ requiring $n - O(\log \log n)$ queries. We give an explicit function on the balanced slice requiring $n - O(\log n)$ queries based on independent sets in Johnson graphs. On the weight-2 slice, we show that hard functions are closely related to Ramsey graphs. Further we describe a simple way of transforming functions on the hypercube to functions on the balanced slice while preserving several complexity measures.