Improved Tree Sparsifiers in Near-Linear Time
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| Format: | Preprint |
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2025
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| _version_ | 1866912905563209728 |
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| author | Agassy, Daniel Dorfman, Dani Kaplan, Haim |
| author_facet | Agassy, Daniel Dorfman, Dani Kaplan, Haim |
| contents | A \emph{tree cut-sparsifier} $T$ of quality $α$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $α$. A \emph{tree flow-sparsifier} $T$ of quality $α$ guarantees that every demand that can be routed in $T$ can also be routed in $G$ with congestion at most $α$.
We present a near-linear time algorithm that, for any undirected capacitated graph $G=(V,E,c)$, constructs a tree cut-sparsifier $T$ of quality $O(\log^{2} n \log\log n)$, where $n=|V|$. This nearly matches the quality of the best known polynomial construction of a tree cut-sparsifier, of quality $O(\log^{1.5} n \log\log n)$ [Räcke and Shah, ESA~2014]. By the flow-cut gap, our result yields a tree flow-sparsifier (and congestion-approximator) of quality $O(\log^{3} n \log\log n)$. This improves on the celebrated result of [Räcke, Shah, and Täubig, SODA~2014] (RST) that gave a near-linear time construction of a tree flow-sparsifier of quality $O(\log^{4} n)$.
Our algorithm builds on a recent \emph{expander decomposition} algorithm by [Agassy, Dorfman, and Kaplan, ICALP~2023], which we use as a black box to obtain a clean and modular foundation for tree cut-sparsifiers. This yields an improved and simplified version of the RST construction for cut-sparsifiers with quality $O(\log^{3} n)$. We then introduce a near-linear time \emph{refinement phase} that controls the load accumulated on boundary edges of the sub-clusters across the levels of the tree. Combining the improved framework with this refinement phase leads to our final $O(\log^{2} n \log\log n)$ tree cut-sparsifier. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Tree Sparsifiers in Near-Linear Time Agassy, Daniel Dorfman, Dani Kaplan, Haim Data Structures and Algorithms A \emph{tree cut-sparsifier} $T$ of quality $α$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $α$. A \emph{tree flow-sparsifier} $T$ of quality $α$ guarantees that every demand that can be routed in $T$ can also be routed in $G$ with congestion at most $α$. We present a near-linear time algorithm that, for any undirected capacitated graph $G=(V,E,c)$, constructs a tree cut-sparsifier $T$ of quality $O(\log^{2} n \log\log n)$, where $n=|V|$. This nearly matches the quality of the best known polynomial construction of a tree cut-sparsifier, of quality $O(\log^{1.5} n \log\log n)$ [Räcke and Shah, ESA~2014]. By the flow-cut gap, our result yields a tree flow-sparsifier (and congestion-approximator) of quality $O(\log^{3} n \log\log n)$. This improves on the celebrated result of [Räcke, Shah, and Täubig, SODA~2014] (RST) that gave a near-linear time construction of a tree flow-sparsifier of quality $O(\log^{4} n)$. Our algorithm builds on a recent \emph{expander decomposition} algorithm by [Agassy, Dorfman, and Kaplan, ICALP~2023], which we use as a black box to obtain a clean and modular foundation for tree cut-sparsifiers. This yields an improved and simplified version of the RST construction for cut-sparsifiers with quality $O(\log^{3} n)$. We then introduce a near-linear time \emph{refinement phase} that controls the load accumulated on boundary edges of the sub-clusters across the levels of the tree. Combining the improved framework with this refinement phase leads to our final $O(\log^{2} n \log\log n)$ tree cut-sparsifier. |
| title | Improved Tree Sparsifiers in Near-Linear Time |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2511.06574 |