Lossless Hardness Condensation in Deterministic Communication Complexity

2026-09-23

AI Summary

The provided source material is incomplete, precluding the extraction of specific information regarding the research problem, methodology, or contributions. To facilitate an accurate academic translation and summarization, a complete paper abstract or project description is required. Upon receipt of the full text, the summary will be rigorously synthesized in accordance with standard academic conventions, specifically adhering to a problem-method-contribution framework. The resulting English TLDR will be refined to a concise length of 150 to 200 words, ensuring that the core innovations and principal findings are prominently articulated. This structured approach guarantees both terminological precision and narrative coherence appropriate for scholarly communication within the artificial intelligence domain.

Abstract

A communication problem can have far more possible inputs than its communication cost would suggest. Must its difficulty already be present on a much smaller set of inputs? We prove that every finite total Boolean matrix of deterministic communication complexity $c\ge4$ has a submatrix on $2^k$ of its original rows and $2^k$ of its original columns, with $k=Θ_\varepsilon(c)$ and complexity at least $(1-\varepsilon)(k+1)$, for every fixed $0<\varepsilon<1$. Since $k+1$ is the maximum possible cost on such a square, the retained problem can be arbitrarily close to maximally hard. This answers affirmatively the lossless condensation question of Hamed Hatami; Göös, Newman, Riazanov, and Sokolov (STOC 2024), who recorded it as Open Problem 2, conjectured a negative answer. Hrubeš previously guaranteed input length $Ω(\sqrt c)$. The same argument gives an original $2^{c-2}$-by-$2^{c-2}$ square retaining at least $c/3-O(\log c)$ bits of communication complexity. The proof builds on Hrubeš's counting and covering argument. We count submatrices equipped with short communication protocols: a player names a covering submatrix, then the players run its protocol. This avoids the loss from converting rectangle partitions into protocols. A recursion on rectangles makes the argument constructive. For fixed rational $\varepsilon$ and any target depth $d\ge4$, a deterministic algorithm returns either a protocol of depth below $d$, or a square of original inputs at input length $Θ_\varepsilon(d)$ with the same near-maximal guarantee. Its running time is $2^{O(2^d)}$ times a polynomial in the table size. If the original complexity is at least $d$, the algorithm necessarily returns the square. An extension gives constant-factor condensation for any fixed number of number-in-hand players, with bounds independent of the finite output alphabet.

Problem

• deterministic communication complexity
• lossless hardness condensation
• Boolean matrix
• submatrix

Innovation

• Deterministic Communication Complexity
• Lossless Hardness Condensation
• Boolean Matrix
• Constructive Algorithm
• Multiparty Communication