Bounding $\varepsilon$-scatter dimension via metric sparsity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909347692412928 |
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| author | Bourneuf, Romain Pilipczuk, Marcin |
| author_facet | Bourneuf, Romain Pilipczuk, Marcin |
| contents | A recent work of Abbasi et al. [FOCS 2023] introduced the notion of $\varepsilon$-scatter dimension of a metric space and showed a general framework for efficient parameterized approximation schemes (so-called EPASes) for a wide range of clustering problems in classes of metric spaces that admit a bound on the $\varepsilon$-scatter dimension. Our main result is such a bound for metrics induced by graphs from any fixed proper minor-closed graph class. The bound is double-exponential in $\varepsilon^{-1}$ and the Hadwiger number of the graph class and is accompanied by a nearly tight lower bound that holds even in graph classes of bounded treewidth.
On the way to the main result, we introduce metric analogs of well-known graph invariants from the theory of sparsity, including generalized coloring numbers and flatness (aka uniform quasi-wideness), and show bounds for these invariants in proper minor-closed graph classes.
Finally, we show the power of newly introduced toolbox by showing a coreset for $k$-Center in any proper minor-closed graph class whose size is polynomial in $k$ (but the exponent of the polynomial depends on the graph class and $\varepsilon^{-1}$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounding $\varepsilon$-scatter dimension via metric sparsity Bourneuf, Romain Pilipczuk, Marcin Discrete Mathematics Data Structures and Algorithms A recent work of Abbasi et al. [FOCS 2023] introduced the notion of $\varepsilon$-scatter dimension of a metric space and showed a general framework for efficient parameterized approximation schemes (so-called EPASes) for a wide range of clustering problems in classes of metric spaces that admit a bound on the $\varepsilon$-scatter dimension. Our main result is such a bound for metrics induced by graphs from any fixed proper minor-closed graph class. The bound is double-exponential in $\varepsilon^{-1}$ and the Hadwiger number of the graph class and is accompanied by a nearly tight lower bound that holds even in graph classes of bounded treewidth. On the way to the main result, we introduce metric analogs of well-known graph invariants from the theory of sparsity, including generalized coloring numbers and flatness (aka uniform quasi-wideness), and show bounds for these invariants in proper minor-closed graph classes. Finally, we show the power of newly introduced toolbox by showing a coreset for $k$-Center in any proper minor-closed graph class whose size is polynomial in $k$ (but the exponent of the polynomial depends on the graph class and $\varepsilon^{-1}$). |
| title | Bounding $\varepsilon$-scatter dimension via metric sparsity |
| topic | Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2410.10191 |