Near-Optimal Directed Low-Diameter Decompositions

TL;DR

The paper addresses directed Low-Diameter Decompositions (LDDs) and pioneers a deep connection to Expander Decompositions, achieving near-optimal loss $O(\\log n \\log \\log n)$ in directed graphs. It introduces a two-step framework: (i) reduce LDD construction to a cost-minimization problem via Multiplicative Weights Update, and (ii) realize this cost minimization through a new $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\)

Abstract

Low Diameter Decompositions (LDDs) are invaluable tools in the design of combinatorial graph algorithms. While historically they have been applied mainly to undirected graphs, in the recent breakthrough for the negative-length Single Source Shortest Path problem, Bernstein, Nanongkai, and Wulff-Nilsen [FOCS '22] extended the use of LDDs to directed graphs for the first time. Specifically, their LDD deletes each edge with probability at most $O(\frac{1}{D} \cdot \log^2 n)$, while ensuring that each strongly connected component in the remaining graph has a (weak) diameter of at most $D$. In this work, we make further advancements in the study of directed LDDs. We reveal a natural and intuitive (in hindsight) connection to Expander Decompositions, and leveraging this connection along with additional techniques, we establish the existence of an LDD with an edge-cutting probability of $O(\frac{1}{D} \cdot \log n \log\log n)$. This improves the previous bound by nearly a logarithmic factor and closely approaches the lower bound of $Ω(\frac{1}{D} \cdot \log n)$. With significantly more technical effort, we also develop two efficient algorithms for computing our LDDs: a deterministic algorithm that runs in time $\tilde O(m \cdot poly(D))$ and a randomized algorithm that runs in near-linear time $\tilde O(m)$. We believe that our work provides a solid conceptual and technical foundation for future research relying on directed LDDs, which will undoubtedly follow soon.

Theorems & Definitions (49)

• Definition 1.0: Directed Low-Diameter Decomposition
• Theorem 2.1
• Lemma 2.1: Multiplicative Weight Update
• proof : Proof Sketch.
• Lemma 2.2: Directed Expander Decomposition
• Lemma 2.3
• Lemma 2.3: Lopsided Expander Decomposition
• proof : Proof Sketch.
• Lemma 2.3
• proof : Proof Sketch.