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Structure in Communication Complexity and Constant-Cost Complexity Classes

Hamed Hatami, Pooya Hatami

TL;DR

This survey addresses how matrix-structure parameters such as rank, γ2-norm, discrepancy, and sign-rank govern various communication models, and it emphasizes the reverse problem of characterizing Boolean matrices by these parameters. It develops and surveys a suite of structural notions—blocky matrices, γ2-factorization, and equality-oracle protocols—and connects them to constant-cost complexity classes, Schur multipliers, and harmonic-analysis concepts. Key contributions include analytic lower bounds via γ2 and μ-norms, a precise link between equality-oracle protocols and blocky-rank, counterexamples that separate blocky-rank from log-rank-like conjectures, and a nuanced landscape of probabilistic models (BPP, RP, UP P, PPcc) with explicit separations and open questions. The work highlights significant gaps between known lower-bound techniques and the structures that yield efficient randomized or unbounded-error protocols, signaling a path toward a deeper, more unified theory of structure in communication complexity with potential broad impact on related areas of theory and practice.

Abstract

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often imply a bound on some matrix parameter. The challenge lies in establishing the reverse direction, which requires understanding the structure of Boolean matrices for which a given matrix parameter is small or large. We will discuss several research directions that align with this overarching theme.

Structure in Communication Complexity and Constant-Cost Complexity Classes

TL;DR

This survey addresses how matrix-structure parameters such as rank, γ2-norm, discrepancy, and sign-rank govern various communication models, and it emphasizes the reverse problem of characterizing Boolean matrices by these parameters. It develops and surveys a suite of structural notions—blocky matrices, γ2-factorization, and equality-oracle protocols—and connects them to constant-cost complexity classes, Schur multipliers, and harmonic-analysis concepts. Key contributions include analytic lower bounds via γ2 and μ-norms, a precise link between equality-oracle protocols and blocky-rank, counterexamples that separate blocky-rank from log-rank-like conjectures, and a nuanced landscape of probabilistic models (BPP, RP, UP P, PPcc) with explicit separations and open questions. The work highlights significant gaps between known lower-bound techniques and the structures that yield efficient randomized or unbounded-error protocols, signaling a path toward a deeper, more unified theory of structure in communication complexity with potential broad impact on related areas of theory and practice.

Abstract

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often imply a bound on some matrix parameter. The challenge lies in establishing the reverse direction, which requires understanding the structure of Boolean matrices for which a given matrix parameter is small or large. We will discuss several research directions that align with this overarching theme.
Paper Structure (26 sections, 19 theorems, 51 equations, 2 figures)

This paper contains 26 sections, 19 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation:
  3. Communication complexity classes:
  4. Constant-cost communication classes:
  5. Monochromatic rectangles:
  6. Deterministic communication complexity and rank
  7. Equality oracles and $\gamma_2$-Factorization norm
  8. An analytic version of the rank lower-bound
  9. What are the Boolean matrices with small $\gamma_2$-norm then?
  10. Equality Oracle Protocols
  11. Analogue of the log-rank conjecture for blocky rank is false
  12. Blocky matrices and Idempotents of Schur Multipliers
  13. Nondeterministic Model and $\Pcc^\NP$
  14. The structural properties of $\NP$:
  15. The complexity class $\Pcc^{\mathsf{NP}}$.
  16. ...and 11 more sections

Key Result

proposition 1

For every $m$, the $\gamma_2$-norm of the $m\times m$ identity matrix $\I_m$ is $1$,

Figures (2)

  • Figure 1: The entry at a row $\mathsf{A}$ and a column $\mathsf{B}$ indicates whether $\mathsf{A}\subseteq \mathsf{B}$ or $\mathsf{A}\not\subseteq \mathsf{B}$. A question mark indicates that the relationship is unknown. The separations in grey entries follow trivially via padding.
  • Figure 2: $\mathsf{A}\rightarrow \mathsf{B}$ indicates $\mathsf{A}\subseteq \mathsf{B}$.

Theorems & Definitions (31)

  • definition 1: Weighted Rectangle Ratio
  • definition 2
  • proposition 1
  • proof
  • definition 3: Blocky matrices
  • proposition 2: Livshits MR1332920
  • proof
  • proposition 3
  • proof
  • theorem 1: Quantitative Cohen's theorem for $\Z_2^n$ MR133397Green_Sanders
  • ...and 21 more