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

A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs

Kolla and Tulsiani [KT07,Kolla11} and Arora, Barak and Steurer [ABS10] introduced the technique of subspace enumeration, which gives approximation algorithms for graph problems such as unique games and small set expansion; the running time of such algorithms is exponential in the threshold-rank of the graph. Guruswami and Sinop [GS11,GS12], and Barak, Raghavendra, and Steurer [BRS11] developed an alternative approach to the design of approximation algorithms for graphs of bounded threshold-rank, based on semidefinite programming relaxations in the Lassere hierarchy and on novel rounding techniques. These algorithms are faster than the ones based on subspace enumeration and work on a broad class of problems. In this paper we develop a third approach to the design of such algorithms. We show, constructively, that graphs of bounded threshold-rank satisfy a weak Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan [FK99] for dense graphs. The existence of efficient approximation algorithms is then a consequence of the regularity lemma, as shown by Frieze and Kannan. Applying our method to the Max Cut problem, we devise an algorithm that is faster than all previous algorithms, and is easier to describe and analyze

Sum-of-squares lower bounds for planted clique

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n, 1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for k ~ sqrt(n). In this paper we study the complexity of the planted clique problem under algorithms from the Sum-of-squares hierarchy. We prove the first average case lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the SOS hierarchy cannot find a planted k-clique unless k > n^{1/2r} (up to logarithmic factors). Thus, for any constant number of rounds planted cliques of size n^{o(1)} cannot be found by this powerful class of algorithms. This is shown via an integrability gap for the natural formulation of maximum clique problem on random graphs for SOS and Lasserre hierarchies, which in turn follow from degree lower bounds for the Positivestellensatz proof system. We follow the usual recipe for such proofs. First, we introduce a natural "dual certificate" (also known as a "vector-solution" or "pseudo-expectation") for the given system of polynomial equations representing the problem for every fixed input graph. Then we show that the matrix associated with this dual certificate is PSD (positive semi-definite) with high probability over the choice of the input graph.This requires the use of certain tools. One is the theory of association schemes, and in particular the eigenspaces and eigenvalues of the Johnson scheme. Another is a combinatorial method we develop to compute (via traces) norm bounds for certain random matrices whose entries are highly dependent; we hope this method will be useful elsewhere

New Dependencies of Hierarchies in Polynomial Optimization

We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams (SA) hierarchies. We prove a collection of dependencies among these hierarchies both for general CPOPs and for optimization problems on the Boolean hypercube. Key results include for the general case that the SONC and SOS hierarchy are polynomially incomparable, while SDSOS is contained in SONC. A direct consequence is the non-existence of a Putinar-like Positivstellensatz for SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent. Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that provides a O(n) degree bound.Comment: 26 pages, 4 figure

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.) Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure

Directed Steiner Tree and the Lasserre Hierarchy

The goal for the Directed Steiner Tree problem is to find a minimum cost tree in a directed graph G=(V,E) that connects all terminals X to a given root r. It is well known that modulo a logarithmic factor it suffices to consider acyclic graphs where the nodes are arranged in L <= log |X| levels. Unfortunately the natural LP formulation has a |X|^(1/2) integrality gap already for 5 levels. We show that for every L, the O(L)-round Lasserre Strengthening of this LP has integrality gap O(L log |X|). This provides a polynomial time |X|^{epsilon}-approximation and a O(log^3 |X|) approximation in O(n^{log |X|) time, matching the best known approximation guarantee obtained by a greedy algorithm of Charikar et al.Comment: 23 pages, 1 figur