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

2024-07-29

arXiv.org

AI Summary

This paper resolves a long-standing open problem in communication complexity: whether there exist constant-cost problems—whose communication complexity is independent of input size—that are not reducible to the $k$-Hamming distance decision problem. The authors construct the first explicit counterexample, refuting the prior conjecture that all constant-cost problems admit deterministic reductions to $k$-Hamming distance. Their core method introduces *$f$-codes*, a novel concept in coding theory, and establishes an equivalence between the existence of $f$-codes and affine structure constraints. Leveraging combinatorial distance mappings, probabilistic analysis, and deterministic reductions, they prove that non-affine $f$-codes do not exist for infinitely many input lengths. Consequently, the class of $k$-Hamming-reducible problems is shown to be a strict subset of constant-cost communication problems. Moreover, the work derives a necessary affine condition for $f$-code existence, establishing—for the first time—a deep, rigorous connection between communication complexity and coding theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 $f : {2, 4, 6} o mathbb{N}$, we say an encoding function $E : {0, 1}^n o {0, 1}^m$ 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: $f(4) = (f(2) + f(6))/2$.

Problem

• Identify constant-cost problems irreducible to k-Hamming Distance
• Relate communication complexity to coding-theoretic f-code conditions
• Prove f must be affine if f-codes exist infinitely

Innovation

• Exhibit constant-cost problems beyond k-Hamming reduction
• Introduce f-codes for Hamming distance transformation
• Prove affine condition for f-codes' existence