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A Quasipolynomial (2 + ε)-Approximation for Planar Sparsest Cut

Anupam Gupta
Jason Li
Philip N. Klein
53rd Annual ACM Symposium on Theory of Computing (STOC'21) (2021)
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The (non-uniform) sparsest cut problem is the following classical graph-partitioning problem: given a ``supply'' graph, and demands on pairs of vertices, delete some subset of supply edges to minimize the ratio of the supply edges cut to the total demand of the pairs separated by this deletion. The sparsest cut problem has been a driver for new algorithmic approaches. Despite much effort, there are only a handful of nontrivial classes of supply graphs for which constant-factor approximations are known. We consider the problem for planar graphs, and give a $(2+\eps)$-approximation algorithm that runs in quasipolynomial time. Our approach defines a new structural decomposition of an optimal solution using a ``radial cuts'' primitive. We combine this decomposition with a Sherali-Adams-style linear programming relaxation of the problem, which we then round. This should be compared with the polynomial-time approximation algorithm of Rao~(1999), which uses the metric linear programming relaxation and $\ell_1$-embeddings, and achieves an $O(\sqrt{\log n})$-approximation in polynomial time.