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The (uniform) directed sparsest cut problem asks for a cut $(S,\bar{S})$ in a directed graph $G=(V,E)$ which minimize the ratio $\frac{\delta_{out}(S) }{|S||\bar{S}|}$, where $\delta_{out}$ is the total capacity of edges coming out from $S$. The best approximation ratio for the problem is $O(\sqrt{\log n})$ via the paper $O(\sqrt{\log n})$ approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems.

The undirected version of the problem on planar graphs have a factor two approximation algorithm: Finding separator cuts in planar graphs within twice the optimal ! Does anybody aware of a similar result for directed sparsest cut on planar graphs ? Can we apply the same technique for the undirected version to get a constant ratio?

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