We give a new (1 + )-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size n/r expand by a factor √ log n log r bigger, for some small r; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-r Lasserre relaxation. The other is combinatorial and involves a new notion called Small Set Expander Flows (inspired by the expander flows of

more »

... 9]) which we show exists in the input graph. Both algorithms run in time 2 O(r) poly(n). We also show similar approximation algorithms in graphs with genus g with an analogous local expansion condition. This is the first algorithm we know of that achieves (1 + )-approximation on such general family of graphs. 1 We also know how to achieve qualitatively similar results as our main result using BRS rounding + ARV ideas applied to Lasserre solutions at the expense of stricter requirements on small set expansion. However, that method seems unable to give better than O(1)-approximation, whereas GS rounding is able to give (1 + ). 2 In fact, the unexpected appearance of Small Set Expansion (SSE) in this setting is believed to not be a fluke. It appears in the SSE conjecture of Raghavendra and Steurer [RS10] (known to imply the UGC), their "Unique games with SSE" conjecture, as well as in the known subexponential algorithms for unique game. Furthermore, attempts to construct difficult examples for known SDP-based algorithms also end up using graphs (such as the noisy