Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs

TL;DR

This paper extends the sensitivity and degree relationship from Boolean to $m$-ary functions, introduces an $m$-ary sensitivity conjecture, and relates it to graph partitions, showing quadratic bounds for prime $p$.

Contribution

It generalizes the sensitivity-degree relationship to $m$-ary functions and formulates the $m$-ary sensitivity conjecture with graph-theoretic implications.

Findings

Proves $s(f) ext{ in } O( ext{deg}(f)^2)$ for $m$-ary functions.

Introduces the $m$-ary sensitivity conjecture as a polynomial bound.

Shows quadratic lower bounds for the $p$-ary sensitivity conjecture for prime $p$.

Abstract

The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang proved the conjecture via an equivalent graph theoretical reformulation due to Gotsman and Linial. We study $m$-ary functions, i.e., functions $f: T^n \rightarrow T$ where $T\subseteq \mathbb{C}$ is a finite alphabet of cardinality $|T| = m $ and extend the notions of degree $\mathrm{deg}(f)$ and sensitivity $s(f)$ to $m$-ary functions and show $s(f)\in O(\mathrm{deg}(f)^2)$. This generalizes results of Nisan and Szegedy. Conversely, we introduce the $m$-ary sensitivity conjecture, claiming a polynomial upper bound for $\mathrm{deg}(f)$ in terms of $s(f)$. Analogously to results of Gotsman and Linial, we provide a formulation of the conjecture in terms of imbalanced partitions of Hamming graphs into low degree subgraphs. Combining this with ideas of Chung, F\"uredi, Graham and Seymour, we show that for any prime $p$ the bound in the $p$-ary sensitivity conjecture has to be at least quadratic: there exist $p$-ary functions $f$ of arbitrarily large degree and $\mathrm{deg}(f)\in \Omega(s(f)^2)$.