Expander Decomposition and Pruning: Faster, Stronger, and Simpler

Understand

We study the problem of graph clustering where the goal is to partition a graph into clusters, i.e.

• disjoint subsets of vertices, such that each cluster is well connected internally while sparsely connected to the rest of the graph.
• In particular, we use a natural bicriteria notion motivated by Kannan, Vempala, and Vetta which we refer to as {\em expander decomposition}.
• Expander decomposition has become one of the building blocks in the design of fast graph algorithms, most notably in the nearly linear time Laplacian solver by Spielman and Teng, and it also has wide applications in practice.