We develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L1 (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes.We obtain an O( √ log n) approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be Θ( √ log n). We also prove various approximate max-flow/minvertex-cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n).These results have a number of applications. We exhibit an O( √ log n) pseudo-approximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of O( √ log opt) where opt is the size of an optimal separator, improving over the previous best bound of O(log opt). Likewise, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k √ log k), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constantfactor approximation for the treewidth; this can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.

We prove that every n n -point metric space of negative type (and, in particular, every n n -point subset of L 1 L_1 ) embeds into a Euclidean space with distortion O ( log ⁡ n ⋅ log ⁡ log ⁡ n ) O(\sqrt {\log n} \cdot \log \log n) , a result which is tight up to the iterated logarithm factor. As a consequence, we obtain the best known polynomial-time approximation algorithm for the Sparsest Cut problem with general demands. If the demand is supported on a subset of size k k , we achieve an approximation ratio of O ( log ⁡ k ⋅ log ⁡ log ⁡ k ) O(\sqrt {\log k}\cdot \log \log k) .

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A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.It has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into R k , and then apply geometric considerations to the embedding.We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n/k and λ k , the kth smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O( √ λ k log k). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result. The √ log k bound is tight, up to constant factors, for the "noisy hypercube" graphs.

We introduce a method for proving lower bounds on the ecacy of semide nite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2 n δ , for some constant δ > 0. This result yields the rst super-polynomial lower bounds on the semide nite extension complexity of any explicit family of polytopes.Our results follow from a general technique for proving lower bounds on the positive semide nite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sumof-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for 3-.

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into R k , and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n / k and λ k , the k th smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k /2 disjoint sets (one of which will have size at most 2 n / k ), each having expansion at most O (√λ k log k ). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result. The √log k bound is tight, up to constant factors, for the “noisy hypercube” graphs.

We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into L p , 1 < p ≤ 2, requires distortion at least k(p − 1) + 1. An immediate corollary is that there exist arbitrarily large n-point sets X ⊆ L 1 such that any D-embedding of X into d 1 requires d ≥ n Ω(1/D 2 ) . This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L 1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].

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