Many combinatorial optimization problems, including maximum weighted matching and maximum independent set, can be approximated within \((1 \pm \epsilon )\) factors in \(\operatorname {poly}(\log n, 1/\epsilon )\) rounds in the \(\mathsf{LOCAL}\) model via network decompositions [Ghaffari, Kuhn, and Maus, STOC 2018]. These approaches, however, require sending messages of unlimited size, so they do not extend to the more realistic \(\mathsf{CONGEST}~\) model, which restricts the message size to be \(O(\log n)\) bits. For example, despite the long line of research devoted to the distributed matching problem, it still remains a major open problem whether an \((1-\epsilon )\) -approximate maximum weighted matching can be computed in \(\operatorname {poly}(\log n, 1/\epsilon )\) rounds in the \(\mathsf{CONGEST}~\) model. In this paper, we develop a generic framework for obtaining \(\operatorname {poly}(\log n, 1/\epsilon )\) -round \((1\pm \epsilon )\) -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 \(\mathsf{CONGEST}~\) 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 \(\mathsf{LOCAL}\) 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.
