通过扩展器分解缩小稀疏网络中 LOCAL–CONGEST 模型的差距
Narrowing the LOCAL$\unicode{x2013}$CONGEST Gaps in Sparse Networks via Expander Decompositions
AI总结：
本文提出一个通用框架，利用分布式扩展器分解，在排除固定子图的稀疏网络中，为多个组合优化问题实现poly(log n, 1/ε)轮(1±ε)近似，并推广了分布式性质测试。
摘要：
许多组合优化问题可以通过网络分解在 LOCAL 模型中以 $\text{poly}(\log n, 1/ε)$ 轮实现 $(1 \pm ε)$ 近似 [Ghaffari, Kuhn, and Maus, STOC 2018]。这些方法需要发送无限大小的消息，因此无法扩展到 CONGEST 模型，该模型将消息大小限制为 $O(\log n)$ 位。在本文中，我们开发了一个通用框架，用于在 CONGEST 模型中排除固定子图的图上，为许多组合优化问题（包括最大加权匹配、最大独立集和相关聚类）提供 $\text{poly}(\log n, 1/ε)$ 轮的 $(1\pm ε)$ 近似算法。这类图涵盖了文献中研究的许多稀疏网络类别，包括平面图、有界亏格图和有界树宽图。此外，我们展示了我们的框架可以应用于为任意在不相交并下封闭的、排除子图的图性质提供高效的分布式性质测试算法，显著推广了之前针对平面性的分布式性质测试算法 [Levi, Medina, and Ron, PODC 2018 & Distributed Computing 2021]。我们的框架使用分布式扩展器分解算法 [Chang and Saranurak, FOCS 2020] 将图分解为高电导的簇。我们证明了任何排除固定子图的图都允许小的边分隔符。利用这一结果，我们证明了在扩展器分解中每个簇中存在一个高度数顶点，这使得簇的整个图拓扑可以被路由到一个顶点。类似于 LOCAL 模型中网络分解的使用，该顶点将能够在簇诱导的子图上执行任何局部计算，并将结果广播到整个簇。
英文摘要：
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.
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