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Near-optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

Sanjeev Khanna, Aaron L. Putterman, Madhu Sudan

TL;DR

This work extends near-linear-size sparsification from graphs to hypergraphs under linear sketching. By introducing random fingerprinting and k-cut strength decompositions, the authors design a linear-sketch framework that recovers (1±ε)-cut sparsifiers with space \\tilde{O}(n r log(m) / ε^2) and sparsifier size \\tilde{O}(n / ε^2), nearly matching known non-sketching bounds. A matching lower bound shows this dependence on arity $r$ and log(m) is tight up to polylog factors, confirming near-optimality. The results yield dynamic streaming and MPC algorithms with improved space and round complexity, and provide a broad set of techniques (fingerprinting, preprocessing, strength-based sampling) that advance hypergraph sparsification in resource-constrained computation models.

Abstract

A $(1 \pm ε)$-sparsifier of a hypergraph $G(V,E)$ is a (weighted) subgraph that preserves the value of every cut to within a $(1 \pm ε)$-factor. It is known that every hypergraph with $n$ vertices admits a $(1 \pm ε)$-sparsifier with $\tilde{O}(n/ε^2)$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a \emph{linear sketch}) over the hyperedges of $G$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $\widetilde{O}(n r \log(m) / ε^2)$ bits which with high probability contains sufficient information to recover a $(1 \pm ε)$ cut-sparsifier with $\tilde{O}(n/ε^2)$ hyperedges for any hypergraph with at most $m$ edges each of which has arity bounded by $r$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $\widetilde{O}(n r^2 \log^4(m) / ε^2)$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $ε\in (0,1)$, one needs $Ω(nr \log(m/n) / \log(n))$ bits to construct a $(1 \pm ε)$-sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $r$ and $\log(m)$.

Near-optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

TL;DR

This work extends near-linear-size sparsification from graphs to hypergraphs under linear sketching. By introducing random fingerprinting and k-cut strength decompositions, the authors design a linear-sketch framework that recovers (1±ε)-cut sparsifiers with space \\tilde{O}(n r log(m) / ε^2) and sparsifier size \\tilde{O}(n / ε^2), nearly matching known non-sketching bounds. A matching lower bound shows this dependence on arity and log(m) is tight up to polylog factors, confirming near-optimality. The results yield dynamic streaming and MPC algorithms with improved space and round complexity, and provide a broad set of techniques (fingerprinting, preprocessing, strength-based sampling) that advance hypergraph sparsification in resource-constrained computation models.

Abstract

A -sparsifier of a hypergraph is a (weighted) subgraph that preserves the value of every cut to within a -factor. It is known that every hypergraph with vertices admits a -sparsifier with hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a \emph{linear sketch}) over the hyperedges of , and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size bits which with high probability contains sufficient information to recover a cut-sparsifier with hyperedges for any hypergraph with at most edges each of which has arity bounded by . This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every , one needs bits to construct a -sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both and .
Paper Structure (39 sections, 26 theorems, 19 equations, 15 algorithms)

This paper contains 39 sections, 26 theorems, 19 equations, 15 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Conclusion
  4. Organization
  5. Detailed Technical Overview
  6. Graph Sparsification and Hypergraph Sparsification via $k$-cuts
  7. Barriers to Finding Low Strength Hyperedges with Linear Sketches
  8. Efficient Recovery using Random Fingerprinting
  9. Strength Decomposition with Linear Sketches
  10. Simple Sparsification Using Strength Decomposition
  11. Improving the Error Accumulation
  12. Preprocessing to Bound Hypergraph Sizes
  13. Lower Bound
  14. Preliminaries
  15. $\ell_0$-samplers and Vertex Incidence Sketches
  16. ...and 24 more sections

Key Result

Theorem 1.1

For any $\epsilon \in (0,1)$, there is a randomized linear sketch of size $\widetilde{O}(nr \log(m) / \epsilon^2)$ bits that given any $n$-vertex unweighted hypergraph $H$ with at most $m$ edges of arity bounded by $r$, allows recovery of a $(1 \pm \epsilon)$-sparsifier of $H$ with high probability.

Theorems & Definitions (151)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.6
  • ...and 141 more