Sensitivity and Hamming graphs

2025-05-13

AI Summary

This work investigates the relationship between the sensitivity of $m$-ary Boolean functions and the structural properties of the Hamming graph $H(n,m)$, with emphasis on how imbalanced $m$-partite partitions of $H(n,m)$ affect lower bounds on sensitivity. Employing combinatorial graph theory, spectral analysis, and polynomial degree–sensitivity correspondence techniques, the authors first refute the strong $m$-ary sensitivity conjecture proposed by Asensio et al., while rigorously proving its weak variant. The key innovation is an explicit construction of an $m$-partite, imbalanced partition of $H(n,m)$ with maximum degree $O(n^{1-1/m})$, substantially improving prior bounds by Tandya and by Potechin–Tsang. This construction yields a polynomial lower bound on sensitivity in terms of polynomial degree: $s(f) = Omega(deg(f)^{1/m})$. The result establishes a new complexity benchmark for high-dimensional Boolean functions and resolves a central open problem in generalized Boolean function sensitivity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For any $mgeq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, Garc'ia-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.

Problem

• Improve imbalanced partition results for Hamming graphs
• Disprove Strong m-ary Sensitivity Conjecture
• Prove weaker m-ary Sensitivity Conjecture

Innovation

• Imbalanced partition of Hamming graphs
• Disproof of Strong m-ary Sensitivity Conjecture
• Polynomial bound on m-ary function sensitivity