Local Cheeger Inequalities and Sparse Cuts

When computing on massive graphs, it is not tractable to iterate over the full graph. Local graph methods aim to analyze properties in a neighborhood of a seed vertex. Graph partitioning is an example of a problem that has a local analog. It attempts to find a sparse cut of a specific volume near a seed vertex. Previous work attempts to approximate the optimal partition through standard, PageRank, and Heat Kernel PageRank random walks. We review these methods and focus on a recent improved approximation result from the Heat Kernel PageRank. Specifically, we show that the result was obtained via a false Local Cheeger Inequality. We present a counterexample, explain the flaw in the proof, obtain weaker, correct results, and cite implications on Hardness of Approximation if the algorithm is able to achieve its approximation ratio.