Approximating Non-Uniform Sparsest Cut via Generalized Spectra

We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any $\epsilon,\delta \in (0,1)$, given cost and demand graphs with edge weights $C, D$ respectively, we can find a set $T\subseteq V$ with $\frac{C(T,V\setminus T)}{D(T,V\setminus T)}$ at most $\frac{1+\epsilon}{\delta}$ times the optimal non-uniform sparsest cut value, in time $2^{r/(\delta\epsilon)}\poly(n)$ provided $\lambda_r \ge \Phi^*/(1-\delta)$. Here $\lambda_r$ is the $r$'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; $C(T,V\setminus T)$ (resp. $D(T,V\setminus T)$) is the weight of edges crossing the $(T,V\setminus T)$ cut in cost (resp. demand) graph and $\Phi^*$ is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the $r$'th smallest eigenvalue of the normalized Laplacian playing the role of $\lambda_r$ in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting $\ell_1$-embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut.