English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the {\em nonuniform sparsest cut} problem, given two undirected graphs $G$and $H$ over the same set of vertices $V$, we want to find a cut $(S,V-S)$ thatminimizes the ratio between the fraction of $G$-edges that are cut and thefraction of $H$-edges that are cut. The ratio (which is at most 1 in an optimalsolution) is called the {\em sparsity} of the cut. In the {\em uniform sparsestcut} problem, $H$ is a clique over $V$. If $G$ is regular, it is possible tofind a solution to the uniform sparsest cut of cost $O(\sqrt{opt})$ in nearlylinear time. Is such an approximation, which we call "Cheege-type"approximation, achievable in the non-uniform case?  We show that the answer is negative, assuming the Unique Games Conjecture,for general H. Furthermore, the Leighton-Rao linear programming relaxation andthe spectral relaxation fail to find such an approximation even if $H$ is aclique over a subset of vertices. Using semidefinite programming, however, wecan find Cheeger-type approximations in polynomial time whenever the adjacencymatrix of $H$ has rank 1. (This includes the cases in which $H$ is a cliqueover a subset of vertices.)

Is Cheeger-type Approximation Possible for Nonuniform Sparsest Cut?