Core Concepts

There is no randomized constant-cost communication problem that is complete for the class of problems with constant-cost randomized protocols (BPP0). Additionally, the k-Hamming Distance problems form an infinite hierarchy within BPP0.

Abstract

The paper proves two main results:
There is no complete problem for the class BPP0 of communication problems that have constant-cost randomized public-coin protocols. This means there is no single randomized constant-cost communication problem Q such that all other BPP0 problems can be computed by a constant-cost deterministic protocol with access to an oracle for Q.
The k-Hamming Distance problems form an infinite hierarchy within BPP0. Specifically, the authors show that for infinitely many constants k, the k-Hamming Distance problem cannot be reduced to the (k-1)-Hamming Distance problem under constant-cost reductions. This separates 1-Hamming Distance from Equality (0-Hamming Distance), and also separates 1-Hamming Distance from 2-Hamming Distance.
The authors introduce a new Ramsey-theoretic lower bound technique that can prove separations against arbitrary oracles in BPP0, overcoming limitations of previous techniques that could only handle the Equality oracle. They also show that the k-Hamming Distance problems have certain "dimension-free" properties that imply the Integer Inner Product problems IIPd cannot be reduced to k-Hamming Distance under unbounded-size BPP reductions.

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Key Insights Distilled From

by Yuting Fang,... at **arxiv.org** 04-02-2024

Deeper Inquiries

In addition to the k-Hamming Distance problem, other natural communication problems known to be in BPP0 include the Equality problem and the Greater-Than problem. The Equality problem involves determining if two strings are equal, while the Greater-Than problem focuses on comparing the values of two numbers. These problems are related to the k-Hamming Distance hierarchy in that they all have constant-cost randomized public-coin protocols. The k-Hamming Distance hierarchy specifically deals with determining the Hamming distance between two strings, with different values of k representing different levels of distance to be considered. The hierarchy shows that certain k-Hamming Distance problems cannot be reduced to others, indicating a level of complexity within this class of problems.

The techniques developed in the paper could potentially be adapted to prove lower bounds against other specific problems in BPP0. By leveraging the concept of stable sets of matrices and the notion of constant-cost reductions, it may be possible to analyze the structure of different communication problems within BPP0 and establish lower bounds against them. The approach of using Ramsey theory to force certain behaviors in communication protocols could be applied to other problems in the class, provided that the necessary conditions and properties are met.

There are some positive results about the structure of problems in BPP0, such as the stability property observed in the context of the Greater-Than problem. The stability of a problem indicates that there is a maximum size for certain subproblems within it, which can provide insights into the complexity and behavior of the problem. However, there is still ongoing research to fully characterize the problems in BPP0 and identify any natural complete problems for restricted subclasses. Further exploration and analysis of the properties and relationships between different problems in BPP0 could lead to a better understanding of this class of communication problems.

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