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.

A $(ϕ,ε)$-expander-decomposition of a graph $G$ (with $n$ vertices and $m$ edges) is a partition of $V$ into clusters $V_1,\ldots,V_k$ with conductance $Φ(G[V_i]) \ge ϕ$, such that there are at most $εm$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized $\tilde{O}(m/ϕ)$ time algorithm for computing a $(ϕ, ϕ\log^2 {n})$-expander decomposition. This improves upon the $(ϕ, ϕ\log^3 {n})$-expander decomposition also obtained in $\tilde{O}(m/ϕ)$ time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal. One crucial component of SW's algorithm is non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW.