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A Dichotomy Theorem for Multi-Pass Streaming CSPs

Yumou Fei, Dor Minzer, Shuo Wang

Abstract

We show a dichotomy result for $p$-pass streaming algorithms for all CSPs and for up to polynomially many passes. More precisely, we prove that for any arity parameter $k$, finite alphabet $Σ$, collection $\mathcal{F}$ of $k$-ary predicates over $Σ$ and any $c\in (0,1)$, there exists $0<s\leq c$ such that: 1. For any $\varepsilon>0$ there is a constant pass, $O_{\varepsilon}(\log n)$-space randomized streaming algorithm solving the promise problem $\text{MaxCSP}(\mathcal{F})[c,s-\varepsilon]$. That is, the algorithm accepts inputs with value at least $c$ with probability at least $2/3$, and rejects inputs with value at most $s-\varepsilon$ with probability at least $2/3$. 2. For all $\varepsilon>0$, any $p$-pass (even randomized) streaming algorithm that solves the promise problem $\text{MaxCSP}(\mathcal{F})[c,s+\varepsilon]$ must use $Ω_{\varepsilon}(n^{1/3}/p)$ space. Our approximation algorithm is based on a certain linear-programming relaxation of the CSP and on a distributed algorithm that approximates its value. This part builds on the works [Yoshida, STOC 2011] and [Saxena, Singer, Sudan, Velusamy, SODA 2025]. For our hardness result we show how to translate an integrality gap of the linear program into a family of hard instances, which we then analyze via studying a related communication complexity problem. That analysis is based on discrete Fourier analysis and builds on a prior work of the authors and on the work [Chou, Golovnev, Sudan, Velusamy, J.ACM 2024].

A Dichotomy Theorem for Multi-Pass Streaming CSPs

Abstract

We show a dichotomy result for -pass streaming algorithms for all CSPs and for up to polynomially many passes. More precisely, we prove that for any arity parameter , finite alphabet , collection of -ary predicates over and any , there exists such that: 1. For any there is a constant pass, -space randomized streaming algorithm solving the promise problem . That is, the algorithm accepts inputs with value at least with probability at least , and rejects inputs with value at most with probability at least . 2. For all , any -pass (even randomized) streaming algorithm that solves the promise problem must use space. Our approximation algorithm is based on a certain linear-programming relaxation of the CSP and on a distributed algorithm that approximates its value. This part builds on the works [Yoshida, STOC 2011] and [Saxena, Singer, Sudan, Velusamy, SODA 2025]. For our hardness result we show how to translate an integrality gap of the linear program into a family of hard instances, which we then analyze via studying a related communication complexity problem. That analysis is based on discrete Fourier analysis and builds on a prior work of the authors and on the work [Chou, Golovnev, Sudan, Velusamy, J.ACM 2024].

Paper Structure

This paper contains 94 sections, 62 theorems, 267 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Constraint Satisfaction Problems
  3. CSPs in the NP World
  4. CSPs in the Streaming World
  5. Dichotomy Theorems for Streaming Algorithms?
  6. Main Result
  7. Examples: DICUT and 2SAT
  8. Discussion: Sublinear Space vs. Sublinear Time
  9. A Rich World of Approximability Hierarchy?
  10. Techniques
  11. The Linear-Programming Relaxation
  12. Approximating the Value of ${\textsc{BasicLP}}_{\mathcal{I}}$ via a Streaming Algorithm
  13. From an Integrality Gap to Communication Complexity
  14. The Distributional Implicit Hidden Partition (DIHP)
  15. The DIHP Lower Bound
  16. ...and 79 more sections

Key Result

Theorem 1.3

For every $k\in\mathbb{N}$, a family $\mathcal{F}$ of $k$-ary predicates and $0\leqslant s<c\leqslant 1$, either the problem $\mathsf{MaxCSP}(\mathcal{F})[c, s]$ admits an $O(\log^3 n)$-space sketching algorithm, or else for all $\varepsilon>0$, the problem $\mathsf{MaxCSP}(\mathcal{F})[c-\varepsilo

Figures (2)

  • Figure 1: The threshold functions $\vartheta_{\mathcal{F}}$ are shown in solid black lines. Red region stands for hardness, while green region corresponds to parameters where efficient multi-pass streaming algorithms exist.
  • Figure 2: relative power of algorithmic models in approximating CSP value

Theorems & Definitions (165)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: CGSV24
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 155 more