A Quasipolynomial $(2+\varepsilon)$-Approximation for Planar Sparsest Cut

TL;DR

This paper presents a quasipolynomial-time algorithm that achieves a near-constant approximation ratio for the sparsest cut problem specifically in planar graphs, improving upon previous approximation bounds.

Contribution

It introduces a novel structural decomposition and combines it with a linear programming relaxation to attain a $(2+ ext{epsilon})$-approximation for planar sparsest cut.

Findings

Achieves a $(2+ ext{epsilon})$-approximation in quasipolynomial time.

Introduces a new structural decomposition using a 'patching' primitive.

Combines decomposition with Sherali-Adams LP relaxation for rounding.

Abstract

The (non-uniform) sparsest cut problem is the following 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. 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 $(2+\varepsilon)$-approximation algorithm that runs in quasipolynomial time. Our approach defines a new structural decomposition of an optimal solution using a "patching" 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.