Query complexity of Boolean functions on slices

11/29/2022

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by   Farzan Byramji, et al.

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We study the deterministic query complexity of Boolean functions on slices of the hypercube. The k^th slice [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 [n]n/2 requiring n - O(loglog 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.