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Streaming Max-Cut in General Metrics

Shaofeng H. -C. Jiang, Pan Peng, Haoze Wang

TL;DR

This work initiates the study of streaming Max-Cut in general metric spaces with distance oracles, addressing insertion-only, sliding-window, and dynamic streams. It introduces a polylogarithmic-space $(1+ε)$-approximation for sliding-window streams by combining a new insertion-only metric sampling/coreset approach with the smooth histogram framework, and proves a smoothness bound for metric Max-Cut by augmenting the objective with a small linear term. A polynomial-space lower bound for dynamic streams is proved, establishing a clear separation from the insertion-only/sliding-window regime and highlighting a qualitative difference from the Euclidean setting. The results demonstrate that general metric assumptions substantially increase streaming complexity for Max-Cut in the dynamic model, while remaining tractable in sliding-window scenarios via the proposed techniques. Overall, the paper advances understanding of how metric structure and update models shape the feasibility of near-optimal streaming approximations for Max-Cut.

Abstract

Max-Cut is a fundamental combinatorial optimization problem that has been studied in various computational settings. In this work, we initiate the study of its streaming complexity in general metric spaces with access to distance oracles. We give a $(1 + ε)$-approximation algorithm for estimating the Max-Cut value sliding-window streams using only poly-logarithmic space. This is the first sliding-window algorithm for Max-Cut even in Euclidean spaces, and it achieves a similar error-space tradeoff as the state-of-the-art insertion-only algorithms in Euclidean settings [Chen, Jiang, Krauthgamer, STOC'23], but without relying on Euclidean structures. In sharp contrast, we prove a polynomial-space lower bound for any $\mathrm{poly}(n)$-approximation in the dynamic streaming setting. This yields a separation from the Euclidean case, where the polylogarithmic-space $(1+ε)$-approximation extends to dynamic streams. On the technical side, our sliding-window algorithm builds on the smooth histogram framework of [Braverman and Ostrovsky, SICOMP'10]. To make this framework applicable, we establish the first smoothness bound for metric Max-Cut. Moreover, we develop a streaming algorithm for metric Max-Cut in insertion-only streams, whose key ingredient is a new metric reservoir sampling technique.

Streaming Max-Cut in General Metrics

TL;DR

This work initiates the study of streaming Max-Cut in general metric spaces with distance oracles, addressing insertion-only, sliding-window, and dynamic streams. It introduces a polylogarithmic-space -approximation for sliding-window streams by combining a new insertion-only metric sampling/coreset approach with the smooth histogram framework, and proves a smoothness bound for metric Max-Cut by augmenting the objective with a small linear term. A polynomial-space lower bound for dynamic streams is proved, establishing a clear separation from the insertion-only/sliding-window regime and highlighting a qualitative difference from the Euclidean setting. The results demonstrate that general metric assumptions substantially increase streaming complexity for Max-Cut in the dynamic model, while remaining tractable in sliding-window scenarios via the proposed techniques. Overall, the paper advances understanding of how metric structure and update models shape the feasibility of near-optimal streaming approximations for Max-Cut.

Abstract

Max-Cut is a fundamental combinatorial optimization problem that has been studied in various computational settings. In this work, we initiate the study of its streaming complexity in general metric spaces with access to distance oracles. We give a -approximation algorithm for estimating the Max-Cut value sliding-window streams using only poly-logarithmic space. This is the first sliding-window algorithm for Max-Cut even in Euclidean spaces, and it achieves a similar error-space tradeoff as the state-of-the-art insertion-only algorithms in Euclidean settings [Chen, Jiang, Krauthgamer, STOC'23], but without relying on Euclidean structures. In sharp contrast, we prove a polynomial-space lower bound for any -approximation in the dynamic streaming setting. This yields a separation from the Euclidean case, where the polylogarithmic-space -approximation extends to dynamic streams. On the technical side, our sliding-window algorithm builds on the smooth histogram framework of [Braverman and Ostrovsky, SICOMP'10]. To make this framework applicable, we establish the first smoothness bound for metric Max-Cut. Moreover, we develop a streaming algorithm for metric Max-Cut in insertion-only streams, whose key ingredient is a new metric reservoir sampling technique.

Paper Structure

This paper contains 27 sections, 16 theorems, 71 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Sliding-Window Algorithms.
  4. Lower Bound for Dynamic Streams.
  5. Technical Overview
  6. Upper Bounds
  7. An Insertion-only Algorithm.
  8. Smoothness of Metric $\operatorname{Max-Cut}$.
  9. The Lower Bound
  10. Related Work
  11. Preliminaries
  12. An Algorithm for Insertion-Only Streams
  13. Some Tools
  14. The Algorithm
  15. The Analysis
  16. ...and 12 more sections

Key Result

Theorem 1.1

There is sliding-window algorithm for metric $\operatorname{Max-Cut}$ that estimates the optimal value for each window within $(1 + \epsilon)$ factor with constant probability under the distance oracle model, using space $\mathop{\mathrm{poly}}\nolimits(\epsilon^{-1}\log(w \Delta))$ where $\Delta$ i

Theorems & Definitions (35)

  • Theorem 1.1: Informal; see \ref{['thm:sliding_window']}
  • Theorem 1.2: Informal; see \ref{['thm:lb']}
  • Definition 2.1: Weighted Set
  • Definition 2.2: $\operatorname{Max-Cut}$ on Weighted Sets
  • Theorem 3.1
  • Lemma 3.2: DBLP:conf/stoc/ChenJK23
  • Definition 3.3: Coreset Har-PeledM04
  • Lemma 3.4: Chen09
  • Lemma 3.5
  • proof
  • ...and 25 more