Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs (2409.16141v1)

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 $\mathrm{deg}(f)\in \Omega(s(f)^2)$.

• The paper presents a polynomial upper bound for m-ary function sensitivity by extending the Boolean function framework.
• It establishes a quadratic lower bound for prime-based functions, confirming a robust sensitivity-degree relationship using graph partitions.
• The study links function complexity to Hamming graph structure, opening new avenues in complexity theory, cryptography, and coding.

Sensitivity of $m$-ary Functions and Low Degree Partitions of Hamming Graphs

The paper "Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs" by Sara Asensio, Ignacio García-Marco, and Kolja Knauer explores the complexity measures of $m$-ary functions and their implications on Hamming graph structures. This paper generalizes several key results from Boolean function sensitivity to the broader context of $m$-ary functions, which are functions defined on finite alphabets of size $m$.

Key Contributions and Results

The paper begins by revisiting the sensitivity conjecture originally formulated by Nisan and Szegedy, which posits a polynomial relationship between the degree and sensitivity of Boolean functions. This conjecture remained unresolved until Huang's confirmation using graph-theoretic concepts related to the $n$-dimensional hypercube.

The authors extend these fundamental results to $m$-ary functions, where $m$ denotes the cardinality of the finite alphabet $T \subseteq \mathbb{C}$. They successfully define and analyze the sensitivity and degree of $m$-ary functions, yielding several significant findings:

1. Polynomial Upper Bound: The authors extend the sensitivity bound for Boolean functions to $m$-ary functions, showing that $s(f) \in O((\deg(f))^2)$. This generalizes Nisan and Szegedy's results to $m$-ary contexts.
2. $m$-ary Sensitivity Conjecture: Parallel to the original sensitivity conjecture, the conjecture for $m$-ary functions claims a polynomial upper bound for the degree in terms of sensitivity. The authors provide a graph-theoretic formulation analogous to Gotsman and Linial's results, implying imbalanced partitions of Hamming graphs into low-degree subgraphs.
3. Quadratic Lower Bound: For any prime $p$, they prove that the bound in the $p$-ary sensitivity conjecture must be at least quadratic. They demonstrate the existence of $p$-ary functions of arbitrarily large degree for which $\deg(f) = \Omega(s(f)^2)$.
4. Implications for Hamming Graphs: By framing the problem in terms of Hamming graphs $H(n,m)$, the authors translate the sensitivity conjecture into a problem involving imbalanced partitions into low-degree subgraphs.

Analytical Techniques

The methodology employed involves a blend of combinatorial techniques and polynomial representation of functions. Key steps include defining and computing the degree of $m$-ary functions using interpolating polynomials, exploring block sensitivity, and leveraging graph-theoretic perspectives to correlate function complexity with graph structural properties.

Implications and Future Directions

The implications of these results are manifold:

• Graph Theory: The findings encourage further exploration into the structural properties of Hamming graphs and similar graph classes, particularly regarding induced subgraphs and their degree properties.
• Complexity Theory: Establishing polynomial relationships between sensitivity and degree across $m$-ary functions bolsters the theoretical underpinnings of complexity measures. This could impact how algorithms are designed and optimized for multi-valued logic systems.
• Cryptography and Coding Theory: Multi-valued logics are foundational in these fields, and understanding their complexity measures can guide the development of more efficient cryptographic protocols and error-correcting codes.

Conclusion

The paper by Asensio, García-Marco, and Knauer makes substantial progress in generalizing sensitivity and degree relationships from Boolean functions to $m$-ary functions, offering new insight into low-degree partitions of Hamming graphs. Their conjectures and findings not only extend existing theoretical constructs but also open new avenues for research at the intersection of function complexity and graph theory. Future research will likely focus on refining these bounds, exploring other graph products, and applying these concepts in practical computational settings.