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

Zander Kelley, Xin Lyu

TL;DR

The paper advances the 3-player NOF landscape by strengthening the density-sifting mechanism for grid norms and replacing the previous two-sided pseudorandomness requirement with a one-sided condition. Central to the work is a structural result: small cylinder intersections can be efficiently covered by a limited number of slice functions, enabling fractional and integral coverings with favorable bounds. This leads to a sharper $\Omega(\log^{1/2}(N))$ lower bound for deterministic (or non-deterministic) 3-NOF and provides a concrete explicit hard function with improved separation from randomized protocols. The results bridge grid-norm analysis, spread-matrix techniques, and pseudorandomness in NOF, and open avenues for extending the framework to $k$-party NOF and hypergraph settings.

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 $f:[N]^3 \rightarrow \{0,1\}$ 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 $Ω(\log^{1/3}(N))$ bits of communication with a deterministic (or non-deterministic) protocol. We show a stronger $Ω(\log^{1/2}(N))$ 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.

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

TL;DR

The paper advances the 3-player NOF landscape by strengthening the density-sifting mechanism for grid norms and replacing the previous two-sided pseudorandomness requirement with a one-sided condition. Central to the work is a structural result: small cylinder intersections can be efficiently covered by a limited number of slice functions, enabling fractional and integral coverings with favorable bounds. This leads to a sharper lower bound for deterministic (or non-deterministic) 3-NOF and provides a concrete explicit hard function with improved separation from randomized protocols. The results bridge grid-norm analysis, spread-matrix techniques, and pseudorandomness in NOF, and open avenues for extending the framework to -party NOF and hypergraph settings.

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 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 bits, but requires bits of communication with a deterministic (or non-deterministic) protocol. We show a stronger 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 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.
Paper Structure (15 sections, 13 theorems, 70 equations)

This paper contains 15 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Cylinder Intersections.
  3. Explicit Separations for Deterministic and Randomized NOF Complexity.
  4. Triangle Removal Lemma.
  5. An Open Question for $k$-party NOF.
  6. Preliminaries
  7. Norms and Flat Norms of Vectors
  8. Norms and Flat Norms of Matrices
  9. Spread Matrices and Their Products
  10. Proof Overview
  11. Spreadness of Matrices
  12. Product of Spread Matrices is Near-Uniform
  13. Proof of the Removal Lemma
  14. Proof of Proposition \ref{['prop:evasive-to-largeness']}
  15. Number-on-Forehead Communication Lower Bound

Key Result

Theorem 1.2

Let $f(x,y),g(x,z),h(y,z)$ be some indicator functions on $\Omega := X \times Y \times Z$, and consider the cylinder intersection Suppose that $F$ is small, i.e. Then, $F$ has a reasonably small (pointwise) cover $F'$, for some slice functions $s_i(x,y,z)$. Specifically,

Theorems & Definitions (24)

  • Definition 1.1: Slice Function
  • Theorem 1.2: Slice Function "Removal Lemma"
  • Theorem 1.3
  • Theorem 1.4: Triangle Removal Lemma
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Lemma 3.2
  • ...and 14 more