Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut

This paper ties the line of work on algorithms that find an O(¿(log n))-approximation to the SPARSEST CUT together with the line of work on algorithms that run in subquadratic time by using only single-commodity flows. We present an algorithm that simultaneously achieves both goals, finding an O(¿(log (n)/¿))-approximation using O(n^¿ log^O(1) n) max-flows. The core of the algorithm is a stronger, algorithmic version of Arora et al.´s structure theorem, where we show that matching-chaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks. By using that specialized algorithm in place of a black-box solver, we are able to solve those instances much more efficiently. We also show the cut-matching game framework can not achieve an approximation any better than ¿(log(n)/log log(n)) without re-routing flow.