Narrowing the LOCAL$\unicode{x2013}$CONGEST Gaps in Sparse Networks via Expander Decompositions

TL;DR

This paper develops a CONGEST-model framework for polylogarithmic-ε approximation of key combinatorial problems on $H$-minor-free graphs by exploiting expander decompositions. The core technique gathers cluster topology to a leader within each high-conductance component, enabling local computation and broadcast akin to LOCAL network decompositions but compatible with $O(\log n)$-bit messages. The authors prove structural results (small edge separators with $O(\sqrt{\Delta n})$ size) and provide a versatile routing scheme that supports both unweighted and weighted problems, including maximum weighted matching, maximum independent set, and correlation clustering, as well as distributed property testing for minor-closed properties closed under disjoint union. They also present a detailed treatment of graph decompositions and a weighted matching algorithm that integrates a distributed scaling framework with expander routing, yielding $\varepsilon^{-O(1)}\log^{O(1)} n$-round algorithms in CONGEST for $H$-minor-free networks. Substantial follow-on work builds on these techniques to achieve deterministic poly$(\log n,1/\varepsilon)$ rounds for general MWMs and refined decomposition/routing algorithms, extending applicability to broader sparse-network classes. The results close a significant gap between LOCAL and CONGEST for a broad family of sparse graphs with practical distributed-performance implications.

Abstract

Many combinatorial optimization problems can be approximated within $(1 \pm ε)$ factors in $\text{poly}(\log n, 1/ε)$ rounds in the LOCAL model via network decompositions [Ghaffari, Kuhn, and Maus, STOC 2018]. These approaches require sending messages of unlimited size, so they do not extend to the CONGEST model, which restricts the message size to be $O(\log n)$ bits. In this paper, we develop a generic framework for obtaining $\text{poly}(\log n, 1/ε)$-round $(1\pm ε)$-approximation algorithms for many combinatorial optimization problems, including maximum weighted matching, maximum independent set, and correlation clustering, in graphs excluding a fixed minor in the CONGEST model. This class of graphs covers many sparse network classes that have been studied in the literature, including planar graphs, bounded-genus graphs, and bounded-treewidth graphs. Furthermore, we show that our framework can be applied to give an efficient distributed property testing algorithm for an arbitrary minor-closed graph property that is closed under taking disjoint union, significantly generalizing the previous distributed property testing algorithm for planarity in [Levi, Medina, and Ron, PODC 2018 & Distributed Computing 2021]. Our framework uses distributed expander decomposition algorithms [Chang and Saranurak, FOCS 2020] to decompose the graph into clusters of high conductance. We show that any graph excluding a fixed minor admits small edge separators. Using this result, we show the existence of a high-degree vertex in each cluster in an expander decomposition, which allows the entire graph topology of the cluster to be routed to a vertex. Similar to the use of network decompositions in the LOCAL model, the vertex will be able to perform any local computation on the subgraph induced by the cluster and broadcast the result over the cluster.

Figures (3)

• Figure 1: An example illustrating \ref{['dfn:freevertex']}, \ref{['dfn:freeblossom']}, and \ref{['dfn:contracted']}. In this example, $\Omega = \{B_1, B_2, B_3\}$ where $B_2$ is nested inside $B_3$. For the free vertex types, $\hat{F} = \{a,g\}$, $\hat{F}_s = \{g \}$ and $\hat{F}_{b} = \{a\}$. For the blossom types, $B_1$ is a free blossom, whereas $B_2$ and $B_3$ are regular blossoms. Lastly, $\hat{V}_1 / \Omega = \{a,b,c,d,e\}$, $\hat{V}_2 / \Omega = \{f,g\}$, and $\hat{V}_3 /\Omega = \emptyset$.
• Figure 2: The procedure $\textsc{Cut}(\hat{V}_1, \ldots, \hat{V}_k)$.
• Figure 3: The modified Duan-Pettie algorithm with distributed expander decompositions

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2
• Lemma 2.1
• proof