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Single-Pass Streaming CSPs via Two-Tier Sampling

Amir Azarmehr, Soheil Behnezhad, Shane Ferrante

Abstract

We study the maximum constraint satisfaction problem, Max-CSP, in the streaming setting. Given $n$ variables, the constraints arrive sequentially in an arbitrary order, with each constraint involving only a small subset of the variables. The objective is to approximate the maximum fraction of constraints that can be satisfied by an optimal assignment in a single pass. The problem admits a trivial near-optimal solution with $O(n)$ space, so the major open problem in the literature has been the best approximation achievable when limiting the space to $o(n)$. The answer to the question above depends heavily on the CSP instance at hand. The integrality gap $α$ of an LP relaxation, known as the BasicLP, plays a central role. In particular, a major conjecture of the area is that in the single-pass streaming setting, for any fixed $\varepsilon > 0$, (i) an $(α-\varepsilon)$-approximation can be achieved with $o(n)$ space, and (ii) any $(α+\varepsilon)$-approximation requires $Ω(n)$ space. In this work, we fully resolve the first side of the conjecture by proving that an $(α- \varepsilon)$-approximation of Max-CSP can indeed be achieved using $n^{1-Ω_\varepsilon(1)}$ space and in a single pass. Given that Max-DiCut is a special case of Max-CSP, our algorithm fully recovers the recent result of [ABFS26, STOC'26] via a completely different algorithm and proof. On a technical level, our algorithm simulates a suitable local algorithm on a reduced graph using a technique that we call *two-tier sampling*: the algorithm combines both edge sampling and vertex sampling to handle high- and low-degree vertices at the same time.

Single-Pass Streaming CSPs via Two-Tier Sampling

Abstract

We study the maximum constraint satisfaction problem, Max-CSP, in the streaming setting. Given variables, the constraints arrive sequentially in an arbitrary order, with each constraint involving only a small subset of the variables. The objective is to approximate the maximum fraction of constraints that can be satisfied by an optimal assignment in a single pass. The problem admits a trivial near-optimal solution with space, so the major open problem in the literature has been the best approximation achievable when limiting the space to . The answer to the question above depends heavily on the CSP instance at hand. The integrality gap of an LP relaxation, known as the BasicLP, plays a central role. In particular, a major conjecture of the area is that in the single-pass streaming setting, for any fixed , (i) an -approximation can be achieved with space, and (ii) any -approximation requires space. In this work, we fully resolve the first side of the conjecture by proving that an -approximation of Max-CSP can indeed be achieved using space and in a single pass. Given that Max-DiCut is a special case of Max-CSP, our algorithm fully recovers the recent result of [ABFS26, STOC'26] via a completely different algorithm and proof. On a technical level, our algorithm simulates a suitable local algorithm on a reduced graph using a technique that we call *two-tier sampling*: the algorithm combines both edge sampling and vertex sampling to handle high- and low-degree vertices at the same time.

Paper Structure

This paper contains 14 sections, 12 theorems, 34 equations, 1 figure, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution:
  3. Technical Overview
  4. Background
  5. Our contribution
  6. Comparison with AzarmehrBFS26.
  7. Preliminaries & Basic Setup
  8. The Constants
  9. Simplifying the Input
  10. Trevisan's Reduction via Offline Two-Tier Subsampling
  11. Streaming Implementation
  12. Analysis of the Streaming Algorithm
  13. Coupling between online and offline algorithms
  14. General Lemmas About Statistics

Key Result

Theorem 1

For an instance $\textsc{CSP}(\mathcal{F})$, let $\alpha_{\mathcal{F}}$ denote the integrality gap of the corresponding BasicLP. Then for any $\varepsilon > 0$, there is a randomized single-pass streaming algorithm that w.h.p. $(\alpha_{\mathcal{F}} - \varepsilon)$-approximates $\textsc{Max-CSP}{(\m

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Conjecture 1: Streaming Dichotomy
  • Theorem 1
  • Definition 3.1: $\textsc{CSP}(\mathcal{F})$
  • Definition 3.2: $\textsc{BasicLP}$
  • Definition 3.3: Integrality Gap $\alpha_{\mathcal{F}}$ of $\textsc{BasicLP}$
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Lemma 3.7: Yoshida11, Theorem 3.1
  • proof
  • ...and 24 more