No Complete Problem for Constant-Cost Randomized Communication

TL;DR

This work proves there is no complete problem for the constant-cost randomized public-coin communication class $\mathsf{BPP}^0$, resolving a central open question about the structure of constant-cost protocols. It introduces a Ramsey-theoretic technique to obtain oracle lower bounds against arbitrary $\mathsf{BPP}^0$ oracles by enforcing permutation-invariance of the oracle queries, and uses this to show that $k$-Hamming Distance problems form an infinite hierarchy within $\mathsf{BPP}^0$. The authors establish that for any $\mathcal{Q}\in\mathsf{BPP}^0$, there exists a constant $k$ with $\mathsf{D}^{\mathcal{Q}}(\mathsf{EHD}_k)=\omega(1)$, and similarly for $\mathsf{THD}_k$, thereby separating $\mathsf{EHD}_k$ from a broad class of problems (including $\mathsf{IIP}_d$). They further develop the idea of two-tally matrices and leverage VC-dimension arguments to separate $k$-Hamming Distance from Integer Inner Product under unbounded-size $\mathsf{BPP}$ reductions, and show that certain reductions can be simulated by Equality queries. Overall, the paper provides a deeper, dimension-free view of the landscape of constant-cost randomness and opens multiple open directions about the limits of $\mathsf{BPP}^0$ and the reach of $k$-Hamming Distance primitives.

Abstract

We prove that the class of communication problems with public-coin randomized constant-cost protocols, called $BPP^0$, does not contain a complete problem. In other words, there is no randomized constant-cost problem $Q \in BPP^0$, such that all other problems $P \in BPP^0$ can be computed by a constant-cost deterministic protocol with access to an oracle for $Q$. We also show that the $k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$. Previously, it was known only that Equality is not complete for $BPP^0$. We introduce a new technique, using Ramsey theory, that can prove lower bounds against arbitrary oracles in $BPP^0$, and more generally, we show that $k$-Hamming Distance matrices cannot be expressed as a Boolean combination of any constant number of matrices which forbid large Greater-Than subproblems.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Informal
• Definition 1.6: $\mathsf{BPP}^0$
• Remark 1.7
• Example 1.8
• Definition 1.9: Communication with Oracle Queries
• Proposition 1.10