The Integrality Gap of the Goemans-Linial SDP Relaxation for Sparsest Cut Is at Least a Constant Multiple of $\surd$Log N

We prove that the integrality gap of the GoemansaLinial semidefinite programming relaxation for the Sparsest Cut Problem is I\textcopyright(alogn) on inputs with n vertices, thus matching the previously best known upper bound (logn)1/2+o(1) up to lower-order factors. This statement is a consequence of the following new isoperimetric-type inequality. Consider the 8-regular graph whose vertex set is the 5-dimensional integer grid a\textcurrency5 and where each vertex (a,b,c,d,e)a a\textcurrency5 is connected to the 8 vertices (aA$\pm$ 1,b,c,d,e), (a,bA$\pm$ 1,c,d,e), (a,b,cA$\pm$ 1,d,eA$\pm$ a), (a,b,c,dA$\pm$ 1,eA$\pm$ b). This graph is known as the Cayley graph of the 5-dimensional discrete Heisenberg group. Given I\textcopyrighta a\textcurrency5, denote the size of its edge boundary in this graph (a.k.a. the horizontal perimeter of I\textcopyright) by textbarahI\textcopyrighttextbar. For ta a, denote by textbaravtI\textcopyrighttextbar the number of (a,b,c,d,e)a a\textcurrency5 such that exactly one of the two vectors (a,b,c,d,e),(a,b,c,d,e+t) is in I\textcopyright. The vertical perimeter of I\textcopyright is defined to be textbaravI\textcopyrighttextbar= aat=1atextbaravtI\textcopyrighttextbar2/t2. We show that every subset I\textcopyrighta a\textcurrency5 satisfies textbaravI\textcopyrighttextbar=O(textbarahI\textcopyrighttextbar). This vertical-versus-horizontal isoperimetric inequality yields the above-stated integrality gap for Sparsest Cut and answers several geometric and analytic questions of independent interest. The theorem stated above is the culmination of a program whose aim is to understand the performance of the GoemansaLinial semidefinite program through the embeddability properties of Heisenberg groups. These investigations have mathematical significance even beyond their established relevance to approximation algorithms and combinatorial optimization. In particular they contribute to a range of mathematical disciplines including functional analysis, geometric group theory, harmonic analysis, sub-Riemannian geometry, geometric measure theory, ergodic theory, group representations, and metric differentiation. This article builds on the above cited works, with the atwista that while those works were equally valid for any finite dimensional Heisenberg group, our result holds for the Heisenberg group of dimension 5 (or higher) but fails for the 3-dimensional Heisenberg group. This insight leads to our core contribution, which is a deduction of an endpoint L1-boundedness of a certain singular integral on a5 from the (local) L2-boundedness of the corresponding singular integral on a3. To do this, we devise a corona-type decomposition of subsets of a Heisenberg group, in the spirit of the construction that David and Semmes performed in an, but with two main conceptual differences (in addition to more technical differences that arise from the peculiarities of the geometry of Heisenberg group). Firstly, theaatomsa of our decomposition are perturbations of intrinsic Lipschitz graphs in the sense of Franchi, Serapioni, and Serra Cassano (plus the requisite awilda regions that satisfy a Carleson packing condition). Secondly, we control the local overlap of our corona decomposition by using quantitative monotonicity rather than Jones-type I$^2$-numbers.