Deterministic Lifting Theorems for One-Way Number-on-Forehead Communication

TL;DR

This paper introduces a deterministic lifting theorem connecting two-party and Number-on-Forehead (NOF) communication models, leading to optimal separations in communication complexity and new proofs for disjointness.

Contribution

The paper presents a novel deterministic lifting theorem for one-way communication, enabling new lower bounds and separations in multi-party NOF communication complexity.

Findings

Optimal explicit separation between randomized and deterministic one-way NOF communication.

Improved separation between one-round and two-round deterministic NOF communication.

New proof establishing $ ext{Ω}(n)$ complexity for three-party set disjointness.

Abstract

Lifting theorems are one of the most powerful tools for proving communication lower bounds, with numerous downstream applications in proof complexity, monotone circuit lower bounds, data structures, and combinatorial optimization. However, to the best of our knowledge, prior lifting theorems have primarily focused on the two-party communication. In this paper, we propose a new lifting theorem that establishes connections between two-party communication and the Number-on-Forehead (NOF) communication model. Specifically, we present a deterministic lifting theorem that translates one-way two-party communication lower bounds into one-way NOF lower bounds. Our lifting theorem yields two applications. First, we obtain an optimal explicit separation between randomized and deterministic one-way NOF communication, even in the multi-player setting. This improves the prior square-root vs. constant separation for three players established by Kelley and Lyu (arXiv 2025). Second, we achieve optimal separations between one-round and two-round deterministic NOF communication, improving upon the previous separation of $\Omega(\frac{n^{1/(k-1)}}{k^k})$ vs. $O(\log n)$ for $k$ players, as shown by Viola and Wigderson (FOCS 2007). Beyond the lifting theorems, we also apply our techniques to the disjointness problem. In particular, we provide a new proof that the deterministic one-way three-party NOF communication complexity of set disjointness is $\Omega(n)$, further demonstrating the broader applicability of our methods.