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Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs

Sanjeev Khanna, Aaron L. Putterman, Madhu Sudan

TL;DR

The paper develops a unified reduction-based framework to preserve cuts across diverse hypergraph models, showing that difficult directed and submodular hypergraph sparsification tasks can be solved by reductions to simpler undirected or symmetric submodular settings. A key contribution is a directed-to-undirected lifting that preserves Laplacian forms, enabling $O(n^2 \log n \log r / \varepsilon^2)$-sized spectral sparsifiers (and $O(n^2 \log n / \varepsilon^2)$ cut sparsifiers) for directed hypergraphs. Complementary lower bounds establish near-tight space complexity for sketching cuts in directed hypergraphs, while a monotone-submodular-to-symmetric reduction yields near-linear sparsifier sizes $\widetilde{O}(n/\varepsilon^2)$, substantially improving previous $O(n^3/\varepsilon^2)$ bounds. Together, these results illustrate near-optimal trade-offs between sparsifier size and sketch complexity and broaden the toolkit for handling hypergraph cut problems in practice.

Abstract

Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.

Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs

TL;DR

The paper develops a unified reduction-based framework to preserve cuts across diverse hypergraph models, showing that difficult directed and submodular hypergraph sparsification tasks can be solved by reductions to simpler undirected or symmetric submodular settings. A key contribution is a directed-to-undirected lifting that preserves Laplacian forms, enabling -sized spectral sparsifiers (and cut sparsifiers) for directed hypergraphs. Complementary lower bounds establish near-tight space complexity for sketching cuts in directed hypergraphs, while a monotone-submodular-to-symmetric reduction yields near-linear sparsifier sizes , substantially improving previous bounds. Together, these results illustrate near-optimal trade-offs between sparsifier size and sketch complexity and broaden the toolkit for handling hypergraph cut problems in practice.

Abstract

Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.
Paper Structure (9 sections, 19 theorems, 35 equations)

This paper contains 9 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Improved Bounds for Directed Hypergraph Sparsification
  3. Lower Bounds for Sketching Cuts in Directed Hypergraphs
  4. Cut Sparsifiers for Monotone Submodular Hypergraphs
  5. Overview
  6. Preliminaries
  7. Directed to Undirected Hypergraph Sparsification
  8. Space Lower-bounds for Sketching Cuts in Directed Hypergraphs
  9. Monotone Hypergraph Sparsifiers

Key Result

Theorem 1.1

For $H = (V, E)$ an a directed hypergraph on $n$ vertices, one can compute an undirected hypergraph $\psi(H)$ on $n^2+1$ vertices in time $O(mr^2)$ (where $m$ is the number of hyperedges in $H$, and $r$ is the maximum size of any hyperedge in $H$), such that for any $x \in \mathbb{R}^n$, one can als Moreover, for any hyperedge $e \in H$, there is a single corresponding hyperedge $\psi(e)$ in $\psi

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 33 more