Rounding Lasserre SDPs using column selection and spectrum-based approximation schemes for graph partitioning and Quadratic IPs

We present an approximation scheme for minimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Sparsest Cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NP-hardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time $n^{O(r/\epsilon^2)}$ with approximation ratio $\frac{1+\epsilon}{\min\{1,\lambda_r\}}$, where $\lambda_r$ is the $r$'th smallest eigenvalue of the normalized graph Laplacian $\mathcal{L}$. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lower-order terms of the stipulated bound. Our results imply $(1+O(\epsilon))$ factor approximation in time $n^{O(r^\ast/\epsilon^2)}$ where is the number of eigenvalues of $\mathcal{L}$ smaller than $1-\epsilon$ (for variants of sparsest cut, $\lambda_{r^\ast} \ge \mathrm{OPT}/\epsilon$ also suffices, and as $\mathrm{OPT}$ is usually $o(1)$ on interesting instances of these problems, this requirement on $r^\ast$ is typically weaker). For Unique Games, we give a factor $(1+\frac{2+\epsilon}{\lambda_r})$ approximation for minimizing the number of unsatisfied constraints in $n^{O(r/\epsilon)}$ time, improving upon an earlier bound for solving Unique Games on expanders. We also give an algorithm for independent sets in graphs that performs well when the Laplacian does not have too many eigenvalues bigger than $1+o(1)$.