Lower Bounds on Flow Sparsifiers with Steiner Nodes
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910021105745920 |
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| author | Chen, Yu Tan, Zihan Yang, Mingyang |
| author_facet | Chen, Yu Tan, Zihan Yang, Mingyang |
| contents | Given a large graph $G$ with a set of its $k$ vertices called terminals, a \emph{quality-$q$ flow sparsifier} is a small graph $G'$ that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor $q\ge 1$, called the \emph{quality}. Constructing flow sparsifiers with good quality and small size ($|V(G')|$) has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in $V(G)$, and then contract each part into a supernode to obtain $G'$. When $G'$ is only allowed to contain all terminals, the best quality is shown to be $O(\log k/\log\log k)$ and $Ω(\sqrt{\log k/\log\log k})$. In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist $k$-terminal graphs such that, even if we allow $k\cdot 2^{(\log k)^{Ω(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $Ω\big((\log k)^{0.3}\big)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lower Bounds on Flow Sparsifiers with Steiner Nodes Chen, Yu Tan, Zihan Yang, Mingyang Data Structures and Algorithms Given a large graph $G$ with a set of its $k$ vertices called terminals, a \emph{quality-$q$ flow sparsifier} is a small graph $G'$ that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor $q\ge 1$, called the \emph{quality}. Constructing flow sparsifiers with good quality and small size ($|V(G')|$) has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in $V(G)$, and then contract each part into a supernode to obtain $G'$. When $G'$ is only allowed to contain all terminals, the best quality is shown to be $O(\log k/\log\log k)$ and $Ω(\sqrt{\log k/\log\log k})$. In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist $k$-terminal graphs such that, even if we allow $k\cdot 2^{(\log k)^{Ω(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $Ω\big((\log k)^{0.3}\big)$. |
| title | Lower Bounds on Flow Sparsifiers with Steiner Nodes |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2602.12645 |