Improved Directed Expander Decompositions

TL;DR

This work extends expander decomposition methods to directed, capacitated graphs, addressing core obstacles that arise when moving from undirected to directed settings. The authors adapt the non-stop cut-matching game, introducing grafting, partial vertex deletions, and column-based matchings to achieve robust progress toward directing both in- and out-expansion certificates. They develop two-flow certifying constructions, trimming, and a dynamic Push-Pull-Relabel framework (augmented with link-cut trees) to handle weights and deletions, overcoming the flow-decomposition barrier for weighted graphs. The resulting framework yields near-linear-time directed expander decompositions for capacitated graphs and weak-expander decompositions with polylogarithmic overhead, improving on prior work and enabling faster directed max-flow and related dynamic problems. Overall, the paper delivers a cohesive set of techniques to extend undirected expander decomposition guarantees to directed, capacitated graphs with practical running times and strong theoretical guarantees.

Abstract

We obtain faster expander decomposition algorithms for directed graphs, matching the guarantees of Saranurak and Wang (SODA 2019) for expander decomposition on undirected graphs. Our algorithms are faster than prior work and also generalize almost losslessly to capacitated graphs. In particular, we obtain the first directed expander decomposition algorithm for capacitated graphs in near-linear time with optimal dependence on $φ$. To obtain our result, we provide the first implementation and analysis of the non-stop cut-matching game for directed, capacitated graphs. All existing directed expander decomposition algorithms instead temporarily add ''fake edges'' before pruning them away in a final cleanup step. Our result shows that the natural undirected approach applies even to directed graphs. The difficulty is in its analysis, which is technical and requires significant modifications from the original setting of undirected graphs.

Figures (3)

• Figure 1: An illustration of a post-processing grafting step with $D$ the set of deleted nodes and $A$ the certified expanding set. The orange flow paths correspond to the embedding of a matching from $D$ to $A$. The purple flow paths correspond to the embedding of a matching from $D$ to $A$ in the reversed graph.
• Figure 2: A graph of the evolution of $\boldsymbol\ell(v)$ over time. The green regions are time intervals $s_i \leq t \leq r_i$ where the $\boldsymbol\ell(v)$ is non-decreasing. The red regions are $r_i \leq t \leq s_{i+1}$ where $\boldsymbol\ell(v)$ is non-increasing. The partition shown is minimal.
• Figure 3: Depiction of the edge-charging argument for a Path $P$ leaving $A_t$ via $e$ and returning via $e'$. $P$ crosses in and out of $C_1$ via $f$ and $f'$, and $1$ is the smallest index of a cut the subpath $P[e,e']$ passes through. Although $P$ also crosses in and out of $C_2$, the edge into $C_2$ starts in $C_1$. Hence we cannot charge to edges crossing in and out of $C_2$ since $C_2$ is only necessarily sparse in $G[V \setminus C_1]$.

Theorems & Definitions (89)

• Theorem 1.1: Uncapacitated Expander Decomposition
• Theorem 1.2: Capacitated Expander Decomposition
• Theorem 1.3: Weak-Expander Decomposition
• Theorem 5.1: Cut-Matching
• Remark 5.2
• Remark 5.3
• Remark 5.4
• Remark 5.5
• Remark 5.6
• Remark 5.7