Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204v2)

Abstract: Every known communication problem whose randomized communication cost is constant (independent of the input size) can be reduced to $k$-Hamming Distance, that is, solved with a constant number of deterministic queries to some $k$-Hamming Distance oracle. We exhibit the first examples of constant-cost problems which cannot be reduced to $k$-Hamming Distance. To prove this separation, we relate it to a natural coding-theoretic question. For $f : {2, 4, 6} \to \mathbb{N}$, we say an encoding function $E : {0, 1}^n \to {0, 1}^m$ is an $f$-code if it transforms Hamming distances according to $\mathrm{dist}(E(x), E(y)) = f(\mathrm{dist}(x, y))$ whenever $f$ is defined. We prove that, if there exist $f$-codes for infinitely many $n$, then $f$ must be affine: $f(4) = (f(2) + f(6))/2$.

• The paper introduces the {4,4$-Hamming Distance problem, demonstrating a constant-cost communication challenge that cannot be reduced to any k-Hamming Distance formulation.
• The paper establishes a new separation in communication complexity by proving that any encoding preserving small Hamming distances must satisfy an affine condition, ruling out simple k-Hamming reductions.
• The paper's findings have practical implications for developing efficient algorithms and protocols in privacy-preserving data analysis and robust machine learning.

Constant-Cost Communication is not Reducible to $k$-Hamming Distance

This paper investigates the boundaries of constant-cost communication in the context of computational complexity, specifically delineating the relationship between various communication problems and the $k$-Hamming Distance problem. Notably, it provides robust theoretical underpinnings to demonstrate that there exist constant-cost communication problems that cannot be reduced to any $k$-Hamming Distance problem.

Overview

The paper's central result asserts the existence of constant-cost communication problems not reducible to $k$-Hamming Distance. This is established through the introduction of a new problem, termed $\{4,4\$-Hamming Distance$$. Unlike prior constant-cost problems, which could be reformulated in terms of$$k$$-Hamming Distance, this new problem exhibits characteristics that defy such a straightforward reduction.</p> <h3 class='paper-heading' id='key-concepts'>Key Concepts</h3> <ol> <li><strong>$$k$$-Hamming Distance</strong>: Counts the number of positions at which the corresponding symbols of two strings differ.</li> <li><strong>Constant-Cost Communication</strong>: Protocols whose communication complexity does not depend on the input size.</li> <li><strong>$$f$$-Codes</strong>: Encoding functions that preserve specific Hamming distances within certain bounds.</li> </ol> <h3 class='paper-heading' id='main-contributions'>Main Contributions</h3> <ul> <li><strong>Invention of$$\{4,4\$-Hamming Distance: The paper describes a constant-cost communication problem where each input consists of two binary matrices, and the output is determined based on whether exactly two rows have a Hamming distance of 4 from their corresponding rows in the other matrix. A random partitioning protocol is designed, showing how this problem can be addressed within the constant-cost framework without using $k$-Hamming Distance.

Separation Proof: It establishes that $\{4,4\$-Hamming Distance attempts to answer a coding-theoretic question about Hamming distance encodings. By proving certain necessary properties of encodings preserving small Hamming distances, the authors show that these properties exclude realizability by any set of $k$-Hamming Distance reductions.

Mathematical Framework

The paper explores the structure of $f$-codes, necessary for understanding the limitations of $k$-Hamming Distance reductions. It proves the following:

• If for a function $f: \{2,4,6\} \to \mathbb{N}$ and an encoding $E: \{0, 1\}^n \to \{0, 1\}^m$ can be constructed that transforms Hamming distances according to $f$, then the function $f$ must be affine.
• This insight is crucial; the affine nature requirement for $f$ demonstrates the infeasibility of encoding a $\{4,4$ distance problem using a simple $k$-Hamming Distance reduction. Thus, this coding-theoretic result implies a strict separation from the $k$-Hamming Distance hierarchy.

Theoretical and Practical Implications

Theoretical Implications:

1. Hierarchy Establishment: The existence of communication problems not reducible to any $k$-Hamming Distance solidifies a hierarchical structure within constant-cost communication problems.
2. New Class Definition: The introduction of distance-$r$ composed functions extends the family of constant-cost problems, capturing a broader class of problems and encouraging the search for further separable problems.

Practical Implications:

1. Algorithmic Applications: Understanding the limitations of $k$-Hamming Distance reductions can shape the development of more efficient algorithms for new classes of communication problems.
2. Privacy and Learning: Extensions to $f$-codes can influence protocols in privacy-preserving data analysis and robust machine learning algorithms under constraints.

Future Work

The paper opens several paths for advancing the understanding of constant-cost communication:

• Explore New Problems: Identifying problems that do not fit neatly into the $k$-Hamming Distance or even the broader generalizations considered in this work.
• Deeper Coding-Theoretic Connections: Further investigation into the properties of $f$-codes can reveal other non-trivial separations and deepen the understanding of encodings.
• Extended Hierarchies: Generalizing the hierarchy identified by this paper could provide new insights into the structure of communication complexity classes.

Conclusion

This paper makes significant strides in revealing the limitations of $k$-Hamming Distance reductions in constant-cost communication. By introducing and analyzing the $\{4,4\$-Hamming Distance, the authors successfully establish a new dimension in communication complexity, demonstrating that not all constant-cost communication problems succumb to existing reduction techniques. This work not only enhances fundamental knowledge but also paves the way for novel approaches and applications in the field of computational complexity and communication protocols.