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Cut Sparsification and Succinct Representation of Submodular Hypergraphs

Yotam Kenneth, Robert Krauthgamer

TL;DR

This work extends cut sparsification to submodular hypergraphs by introducing polynomial-size sparsifiers that approximate all cuts within a factor $1+\epsilon$ and by introducing a spread-based parameter to tighten sparsifier size in finite-spread cases. It also develops a framework for succinct representation, showing that for additive splitting functions one can achieve encodings with near-linear space in $n$, via a deformation approach that replaces large hyperedges with many small ones, while proving that reweighted sparsifiers can impose substantially larger encoding sizes and in some cases are provably space-inefficient. The authors introduce the concept of deformation, provide upper and lower bounds for deformations of common splitting functions, and derive encoding-size lower bounds that separate reweighted sparsifiers from more compact encodings. Together, these results illuminate the trade-offs between sparsification, succinct encoding, and deformation in submodular hypergraphs, with implications for clustering, ML, and higher-order optimization.

Abstract

In cut sparsification, all cuts of a hypergraph $H=(V,E,w)$ are approximated within $1\pmε$ factor by a small hypergraph $H'$. This widely applied method was generalized recently to a setting where the cost of cutting each hyperedge $e$ is provided by a splitting function $g_e: 2^e\to\mathbb{R}_+$. This generalization is called a submodular hypergraph when the functions $\{g_e\}_{e\in E}$ are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work studied the setting where $H'$ is a reweighted sub-hypergraph of $H$, and measured the size of $H'$ by the number of hyperedges in it. In this setting, we present two results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in $n=|V|$ and $ε^{-1}$; (ii) we propose a new parameter, called spread, and use it to obtain smaller sparsifiers in some cases. We also show that for a natural family of splitting functions, relaxing the requirement that $H'$ be a reweighted sub-hypergraph of $H$ yields a substantially smaller encoding of the cuts of $H$ (almost a factor $n$ in the number of bits). This is in contrast to graphs, where the most succinct representation is attained by reweighted subgraphs. A new tool in our construction of succinct representation is the notion of deformation, where a splitting function $g_e$ is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.

Cut Sparsification and Succinct Representation of Submodular Hypergraphs

TL;DR

This work extends cut sparsification to submodular hypergraphs by introducing polynomial-size sparsifiers that approximate all cuts within a factor and by introducing a spread-based parameter to tighten sparsifier size in finite-spread cases. It also develops a framework for succinct representation, showing that for additive splitting functions one can achieve encodings with near-linear space in , via a deformation approach that replaces large hyperedges with many small ones, while proving that reweighted sparsifiers can impose substantially larger encoding sizes and in some cases are provably space-inefficient. The authors introduce the concept of deformation, provide upper and lower bounds for deformations of common splitting functions, and derive encoding-size lower bounds that separate reweighted sparsifiers from more compact encodings. Together, these results illuminate the trade-offs between sparsification, succinct encoding, and deformation in submodular hypergraphs, with implications for clustering, ML, and higher-order optimization.

Abstract

In cut sparsification, all cuts of a hypergraph are approximated within factor by a small hypergraph . This widely applied method was generalized recently to a setting where the cost of cutting each hyperedge is provided by a splitting function . This generalization is called a submodular hypergraph when the functions are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work studied the setting where is a reweighted sub-hypergraph of , and measured the size of by the number of hyperedges in it. In this setting, we present two results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in and ; (ii) we propose a new parameter, called spread, and use it to obtain smaller sparsifiers in some cases. We also show that for a natural family of splitting functions, relaxing the requirement that be a reweighted sub-hypergraph of yields a substantially smaller encoding of the cuts of (almost a factor in the number of bits). This is in contrast to graphs, where the most succinct representation is attained by reweighted subgraphs. A new tool in our construction of succinct representation is the notion of deformation, where a splitting function is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.
Paper Structure (28 sections, 29 theorems, 117 equations, 1 figure, 1 table)

This paper contains 28 sections, 29 theorems, 117 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Sparsification: All Submodular Hypergraphs
  3. Related Work.
  4. Techniques.
  5. Sparsification: Parameterized by Spread
  6. Techniques.
  7. Succinct Representation
  8. Techniques.
  9. Deformation Lower Bounds
  10. Techniques.
  11. Related Work
  12. Concluding Remarks
  13. Polynomial-Size Sparsifiers for Submodular Hypergraphs
  14. Monotone Submodular Hypergraphs
  15. Sparsifiers for Finite-Spread Splitting Functions
  16. ...and 13 more sections

Key Result

Theorem 1.4

Every submodular hypergraph admits a $(1+\epsilon)$-sparsifier of size $O( \epsilon^{-2} n^3 )$, which is in fact a reweighted sub-hypergraph.

Figures (1)

  • Figure 1: Sparsification bounds for various families of submodular functions, omitting for simplicity $\mathop{\mathrm{poly}}\nolimits(\epsilon^{-1}\log n)$ factors.

Theorems & Definitions (76)

  • Definition 1.1: Sparsifier
  • Theorem 1.4
  • Definition 1.5: Spread
  • Theorem 1.6: Sparsifier Parameterized by Spread
  • Corollary 1.7
  • Definition 1.8
  • Theorem 1.9: Deformation of Additive Functions
  • Corollary 1.10: Additive Functions admit Small Representation
  • Theorem 1.11: Reweighted Sparsifiers Require $\Omega( n^2)$ Bits
  • Theorem 1.12
  • ...and 66 more