Improved Tree Sparsifiers in Near-Linear Time

TL;DR

This work addresses constructing a single-tree sparsifier that preserves all cuts of an undirected capacitated graph within a polylog factor, achieving near-linear preprocessing time. The authors develop a modular framework anchored by expander-decomposition tools and a demand-state formalism to certify cut-sparsifiers, sidestepping explicit multicommodity routing. They first obtain a tree cut-sparsifier with quality $O(\log^3 n)$ and then introduce a near-linear time refinement phase to reach $O(\log^2 n\ log\log n)$, while also deriving a tree flow-sparsifier of quality $O(\log^3 n\ log\log n)$ via the flow-cut gap. The approach hinges on balanced-cut-or-expander primitives and fair-cuts routing to boundary edges, enabling controlled mass transfer with bounded load across levels. This yields practical, scalable sparsifiers that improve on prior near-linear-time bounds and connect modern sparse-cut tools with hierarchical tree constructions.

Abstract

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.

Figures (8)

• Figure 1: (a) Illustration of the first partitioning step, where a cluster $S$ is divided into smaller clusters $Z_1, \ldots, Z_z$. (b) Illustration of the second partitioning step, where the cluster $S$ is first split into two disjoint clusters $L$ and $R$ (shown in green), and then each of $L$ and $R$ is further subdivided according to the $Z$-partition into sub-clusters $L_1, \ldots, L_\ell$ and $R_1, \ldots, R_r$ (shown in blue). (c) The corresponding hierarchical-tree view, where $S$ has two children $L$ and $R$, each of which splits into its second-level sub-clusters.
• Figure 2: Partitioning of cluster $S$. Figures (a)-(c) represent $G$, figures (d)-(f) represent the split graph. Black ovals depict vertices and squares depict split nodes. Figures $(b)$ and $(e)$ correspond to the first partition. Blue edges (and blue squares) depict inter-cluster edges of the first partition (that is, $F$ and $X_F$, respectively). Figure $(f)$ depicts the second partition (which is computed in $G'[S']$). The core $L$ is enveloped in gold. Split nodes in $X_Y$ are filled with gold. Split nodes in $X_{F\cap Y}$ are colored in blue and gold.
• Figure 3: Figure (a) - a cut $(B,W)$ in $G$. Figure (b) - the corresponding cut $(B',W')$ in $G'$.
• Figure 4: Visualization of the sets $L,R,B^R, Y,$ and $\tilde{Y}$ in a cluster $S$. The core $L$ is the region outlined in blue. Red edges correspond to $B^R$, and golden edges correspond to $Y$. The golden-red edge lies in $B^R\cap Y$. The set $\tilde{Y}$ is defined as $\tilde{Y} \mathrel{\mathop:}= Y \setminus B^R$.
• Figure 5: The two-step partition of a cluster $S$. The merge phase (Theorem \ref{['theorem:merge_phase']}) is shown in Figure (a). The refinement phase (Theorem \ref{['theorem:refinement_phase']}) is then applied to the sub-clusters, as shown in Figure (b). The corresponding hierarchical tree is given in Figure (c). Note that the merge phase contributes two levels to the hierarchical tree due to the partition into $L$ (the core) and $R$.

Theorems & Definitions (85)

• Theorem 1.1
• Definition 2.1: Multi-Commodity Flow Problem, Demand Matrix
• Definition 2.2: All-to-all flow problem
• Definition 2.3: Congestion
• Definition 2.4: Cut-Sparsifier
• Definition 2.5: Flow-Sparsifier
• Definition 2.6: Respecting Cut
• Theorem 2.8
• Theorem 2.9
• Theorem 2.10: Flow-Cut Gap aumann1998log