Improved Tree Sparsifiers in Near-Linear Time

A \emph{tree cut-sparsifier} $T$ of quality $\alpha$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $\alpha$. A \emph{tree flow-sparsifier} $T$ of quality $\alpha$ guarantees that every demand that can be routed in $T$ can also be routed in $G$ with congestion at most $\alpha$. 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 $O(\log^{1.5} n \log\log n)$ [R\"acke 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 [R\"acke, Shah, and T\"aubig, 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.