CS 294 Lecture 14: ARV Analysis, Part 3

In which we complete the analysis of the ARV rounding algorithm

We are finally going to complete the analysis of the Arora-Rao-Vazirani rounding algorithm, which rounds a Semidefinite Programming solution of a relaxation of sparsest cut into an actual cut, with an approximation ratio {O(\sqrt {\log |V|})}.

In previous lectures, we reduced the analysis of the algorithm to the following claim.

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CS294 Lecture 13: ARV Analysis, cont’d

In which we continue the analysis of the ARV rounding algorithm

We are continuing the analysis of the Arora-Rao-Vazirani rounding algorithm, which rounds a Semidefinite Programming solution of a relaxation of sparsest cut into an actual cut, with an approximation ratio {O(\sqrt {\log |V|})}.

In previous lectures, we reduced the analysis of the algorithm to the following claim.

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CS359G Lecture 14: ARV Rounding

In which we begin to discuss the Arora-Rao-Vazirani rounding procedure.

Recall that, in a graph {G=(V,E)} with adjacency matrix {A}, then ARV relaxation of the sparsest cut problem is the following semidefinite program.

\displaystyle  \begin{array}{lll} {\rm minimize} & \frac 1 {2|E|} \sum_{u,v} A_{u,v} || {\bf x}_u - {\bf x}_v ||^2\\ {\rm subject\ to}\\ & \sum_{u,v} || {\bf x}_u - {\bf x}_v ||^2 = {|V|^2}\\ & || {\bf x}_u - {\bf x}_v||^2 \leq || {\bf x}_u - {\bf x}_w||^2 + || {\bf x}_w - {\bf x}_v||^2 & \forall u,v,w \in V\\ & \mbox{} {\bf x}_u \in {\mathbb R}^d & \forall u\in V \end{array}

If we denote by {ARV(G)} the optimum of the relaxation, then we claimed that

\displaystyle  ARV(G) \leq \phi(G) \leq O(\sqrt{\log |V|}) \cdot ARV(G)

where the first inequality follows from the fact that {ARV(G)} is a relaxation of {\phi(G)}, and the second inequality is the result whose proof we begin to discuss today.

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