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Space Complexity of Minimum Cut Problems in Single-Pass Streams

Matthew Ding, Alexandro Garces, Jason Li, Honghao Lin, Jelani Nelson, Vihan Shah, David P. Woodruff

TL;DR

This work advances streaming graph algorithms by delivering near-optimal space bounds for minimum cut problems in single-pass streams, across adversarial and random-order settings, and for both approximate and exact objectives. The authors introduce a novel for-each spectral sparsifier in streaming, built from graphical spectral sketches and online leverage-score sampling, achieving $\tilde{O}(n/\varepsilon)$ space and enabling near-linear-time processing with $\tilde{O}(m)$ stream time. This sparsification framework yields a $\tilde{O}(n/\varepsilon)$-space algorithm that approximates minimum cuts and all-pairs effective resistances, plus a random-order one-pass algorithm that computes the exact minimum cut in $\tilde{O}(n)$ space. Complementary lower bounds show near-tight space requirements for both randomized and deterministic one-pass insertion-only settings, and the paper also extends results to random-order streams where exact min-cut is obtainable with optimal space, highlighting a clear separation from adversarial models. The practical impact lies in enabling robust, memory-efficient minimum-cut computations on massive graphs in streaming contexts, with implications for network reliability and related metrics.

Abstract

We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an $\widetilde{O}(n/\varepsilon)$ space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the $Ω(n/\varepsilon^2)$ space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of $\widetilde{O}(n/\varepsilon)$ for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is $\widetilde{O}(1)$, provided that the number of edges in the input graph is at least $(n/\varepsilon^2)^{1+o(1)}$. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the {\it exact} minimum cut on a simple, unweighted graph using $\widetilde{O}(n)$ space. Finally, we give an $Ω(n/\varepsilon^2)$ space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.

Space Complexity of Minimum Cut Problems in Single-Pass Streams

TL;DR

This work advances streaming graph algorithms by delivering near-optimal space bounds for minimum cut problems in single-pass streams, across adversarial and random-order settings, and for both approximate and exact objectives. The authors introduce a novel for-each spectral sparsifier in streaming, built from graphical spectral sketches and online leverage-score sampling, achieving space and enabling near-linear-time processing with stream time. This sparsification framework yields a -space algorithm that approximates minimum cuts and all-pairs effective resistances, plus a random-order one-pass algorithm that computes the exact minimum cut in space. Complementary lower bounds show near-tight space requirements for both randomized and deterministic one-pass insertion-only settings, and the paper also extends results to random-order streams where exact min-cut is obtainable with optimal space, highlighting a clear separation from adversarial models. The practical impact lies in enabling robust, memory-efficient minimum-cut computations on massive graphs in streaming contexts, with implications for network reliability and related metrics.

Abstract

We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is , provided that the number of edges in the input graph is at least . We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the {\it exact} minimum cut on a simple, unweighted graph using space. Finally, we give an space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.

Paper Structure

This paper contains 39 sections, 21 theorems, 9 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Adversarial Streams
  3. Random-order streams
  4. Preliminaries
  5. All-Pairs Effective Resistances
  6. Short-Cycle Decomposition
  7. Overview of Results
  8. Our Techniques
  9. Approximate Minimum Cut.
  10. For-Each Spectral Sparsifier.
  11. Approximate All-Pairs Effective Resistance.
  12. Exact Minimum Cut in Random-order Streams.
  13. Faster Runtime.
  14. Open Problems
  15. Spectral Sparsification and Minimum Cut in Worst-Case Streams
  16. ...and 24 more sections

Key Result

Lemma 1.4

Given an input graph $G$ in a stream, there exists a one-pass streaming algorithm that outputs a subgraph $H$ of $G$ such that with high probability $H$ is a $(1 + \varepsilon)$-spectral sparsifier of $G$. Moreover, the algorithm uses $\widetilde{O}(n/\varepsilon^2)$ space and $\widetilde{O}(m + n/\

Figures (2)

  • Figure 1: Construction of graph $H^+$ for exact minimum cut from Zelke2011Intractability. Red nodes are the ones that are added in addition to the original graph.
  • Figure 2: Construction of graph $H^+(\varepsilon)$ for approximate minimum cut. Red nodes are the ones that are added in addition to the original graph. $G_i$ represents the graph containing (potential) edge $ab$, and $G_0$ and $G_1$ represent the remaining graphs. The double arrow edges represent bicliques between graph $G_0\in G_S$ and $S$, and graph $G_1 \in G_T$ and $T$.

Theorems & Definitions (37)

  • Definition 1.1: For-All Cut Sparsifier
  • Definition 1.2: For-All Spectral Sparsifier
  • Definition 1.3: Graphical For-Each Spectral Sparsifier
  • Lemma 1.4: spectral_stream_2020
  • Definition 2.1: Short-Cycle Decomposition
  • Claim 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Theorem 3.4
  • ...and 27 more