More efficient sifting for grid norms, and applications to multiparty communication complexity

2025-05-02

AI Summary

This work addresses the deterministic versus randomized communication complexity separation problem in the three-party number-on-the-forehead (NOF) model. We present the first explicit construction achieving a deterministic lower bound of Ω(log¹ᐟ² N), improving upon the prior best explicit bound of Ω(log¹ᐟ³ N). Our approach introduces a weaker, one-sided pseudorandomness condition—replacing the previous strong two-sided assumption—and establishes a structural “slicing” theorem for cylinder intersections, enabling efficient covering of small cylinder sets. The construction integrates grid-norm analysis, sifting techniques, triangle counting in tripartite graphs, and pseudorandomness theory. It refines the Kelley–Lovett–Meka framework by substantially relaxing its pseudorandomness requirements and, for the first time, achieves this separation strength explicitly in the three-party NOF model.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Building on the techniques behind the recent progress on the 3-term arithmetic progression problem [KM'23], Kelley, Lovett, and Meka [KLM'24] constructed the first explicit 3-player function $f:[N]^3 ightarrow {0,1}$ that demonstrates a strong separation between randomized and (non-)deterministic NOF communication complexity. Specifically, their hard function can be solved by a randomized protocol sending $O(1)$ bits, but requires $Omega(log^{1/3}(N))$ bits of communication with a deterministic (or non-deterministic) protocol. We show a stronger $Omega(log^{1/2}(N))$ lower bound for their construction. To achieve this, the key technical advancement is an improvement to the sifting argument for grid norms of (somewhat dense) bipartite graphs. In addition to quantitative improvement, we qualitatively improve over [KLM'24] by relaxing the hardness condition: while [KLM'24] proved their lower bound for any function $f$ that satisfies a strong two-sided pseudorandom condition, we show that a weak one-sided condition suffices. This is achieved by a new structural result for cylinder intersections (or, in graph-theoretic language, the set of triangles induced from a tripartite graph), showing that any small cylinder intersection can be efficiently covered by a sum of simple ``slice'' functions.

Problem

• Improving sifting techniques for grid norms
• Strengthening lower bounds in communication complexity
• Relaxing hardness conditions for pseudorandom functions

Innovation

• Improved sifting argument for grid norms
• Relaxed hardness condition for lower bounds
• New structural result for cylinder intersections