Cut Sparsification and Succinct Representation of Submodular Hypergraphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917591428104192 |
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| author | Kenneth, Yotam Krauthgamer, Robert |
| author_facet | Kenneth, Yotam Krauthgamer, Robert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09110 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cut Sparsification and Succinct Representation of Submodular Hypergraphs Kenneth, Yotam Krauthgamer, Robert Data Structures and Algorithms 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. |
| title | Cut Sparsification and Succinct Representation of Submodular Hypergraphs |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2307.09110 |