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Improved Tree Sparsifiers in Near-Linear Time

Authors Daniel Agassy , Dani Dorfman , Haim Kaplan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Graph algorithms analysis
Keywords
  • Tree sparsifiers
  • cut sparsifiers
  • flow sparsifiers
  • congestion approximators
  • expander decomposition
  • near-linear time algorithms

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Abstract

A 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 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(log² 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(log³ 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(log⁴ n).
Our algorithm builds on a recent 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(log³ n). We then introduce a near-linear time 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(log² n log log n) tree cut-sparsifier.

Cite As Get BibTex

Daniel Agassy, Dani Dorfman, and Haim Kaplan. Improved Tree Sparsifiers in Near-Linear Time. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.7

Author Details

Daniel Agassy
  • Tel Aviv University, Israel
Dani Dorfman
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Haim Kaplan
  • Tel Aviv University, Israel

Funding

This work was supported in part by Israel Science Foundation grant no. 1156-23 and the Blavatnik Family Foundation.

References

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