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.Comment: 16 page

Integrality gaps of semidefinite programs for Vertex Cover and relations to ℓ1\ell_1 embeddability of Negative Type metrics

We study various SDP formulations for {\sc Vertex Cover} by adding different constraints to the standard formulation. We show that {\sc Vertex Cover} cannot be approximated better than $2-o(1)$ even when we add the so called pentagonal inequality constraints to the standard SDP formulation, en route answering an open question of Karakostas~\cite{Karakostas}. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution is an $\ell_1$ metric, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar \cite{Char02} to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than 8/7-\eps. To this end we prove a new isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof of (current) Theorem

Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1

Goemans showed that any n points x_1,..., x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an l_{2}^{2}-to-l_{1} embedding with average distortion at most the stable rank, sr(M), of the matrix M consisting of columns {x_i-x_j}_{i<j}. Average distortion embedding suffices for applications such as the SPARSEST CUT problem. Our embedding gives an approximation algorithm for the SPARSEST CUT problem on low threshold-rank graphs, where earlier work was inspired by Lasserre SDP hierarchy, and improves on a previous result of the first and third author [Deshpande and Venkat, in Proc. 17th APPROX, 2014]. Our ideas give a new perspective on l_{2}^{2} metric, an alternate proof of Goemans\u27 theorem, and a simpler proof for average distortion sqrt{d}

Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time $n^{O(k)}$, where $k$ is the treewidth of the graph. This improves on the previous $2^{2^k}$-approximation in time \poly(n) 2^{O(k)} due to Chlamt\'a\v{c} et al. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a $\rho$-approximation for series-parallel graphs (where $\rho \geq 1$), then the Max Cut problem has an algorithm with approximation factor arbitrarily close to $1/\rho$. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than $17/16 - \epsilon$ for $\epsilon > 0$; assuming the Unique Games Conjecture the hardness becomes $1/\alpha_{GW} - \epsilon$. For graphs with large (but constant) treewidth, we show a hardness result of $2 - \epsilon$ assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation

Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low Dimensional Spaces

We consider the problem of embedding a finite set of points x_1, ...x_n in R^d that satisfy l_2^2 triangle inequalities into l_1, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of Magen and Moharammi (2008) ) showed that such points residing in exactly d dimensions can be embedded into l_1 with distortion at most sqrt{d}. We prove the following robust analogue of this statement: if there exists a r-dimensional subspace Pi such that the projections onto this subspace satisfy sum_{i,j in [n]} norm{Pi x_i - Pi x_j}_2^2 >= Omega(1) * sum_{i,j in [n]} norm{x_i - x_j}_2^2, then there is an embedding of the points into l_1 with O(sqrt{r}) average distortion. A consequence of this result is that the integrality gap of the well-known Goemans-Linial SDP relaxation for the Uniform Sparsest Cut problem is O(sqrt{r}) on graphs G whose r-th smallest normalized eigenvalue of the Laplacian satisfies lambda_r(G)/n >= Omega(1)*Phi_{SDP}(G). Our result improves upon the previously known bound of O(r) on the average distortion, and the integrality gap of the Goemans-Linial SDP under the same preconditions, proven in [Deshpande and Venkat, 2014], and [Deshpande, Harsha and Venkat 2016]

Integrality gaps of semidefinite programs for Vertex Cover and relations to ell1_1 embeddability of negative type metrics

We study various SDP formulations for Vertex Cover by adding different constraints to the standard formulation. We rule out approximations better than even when we add the so-called pentagonal inequality constraints to the standard SDP formulation, and thus almost meet the best upper bound known due to Karakostas, of . We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution embeds into &#8467;1 with no distortion, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being &#8467;1 embeddable, the integrality gap is 2&#8201;&#8722;&#8201;o(1). Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar to provide a family of simple examples for negative type metrics that cannot be embedded into &#8467;1 with distortion better than 8/7&#8201;&#8722;&#8201;&#949;. To this end we prove a new isoperimetric inequality for the hypercube. </div