Query complexity of Boolean functions on slices

Farzan Byramji

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arXiv · 2211.16402

Query complexity of Boolean functions on slices

Farzan Byramji

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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.

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Farzan Byramji. 2022-11-29. Query complexity of Boolean functions on slices. https://arxiv.org/abs/2211.16402

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