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- W1585655309 abstract "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 $(S,V-S)$ that minimizes the ratio between the fraction of $G$-edges that are cut and the fraction of $H$-edges that are cut. The ratio (which is at most 1 in an optimal solution) is called the {em sparsity} of the cut. In the {em uniform sparsest cut} problem, $H$ is a clique over $V$. If $G$ is regular, it is possible to find a solution to the uniform sparsest cut of cost $O(sqrt{opt})$ in nearly linear 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 and the spectral relaxation fail to find such an approximation even if $H$ is a clique over a subset of vertices. Using semidefinite programming, however, we can find Cheeger-type approximations in polynomial time whenever the adjacency matrix of $H$ has rank 1. (This includes the cases in which $H$ is a clique over a subset of vertices.)" @default.
- W1585655309 created "2016-06-24" @default.
- W1585655309 creator A5057752375 @default.
- W1585655309 date "2013-03-12" @default.
- W1585655309 modified "2023-09-27" @default.
- W1585655309 title "Is Cheeger-type Approximation Possible for Nonuniform Sparsest Cut?" @default.
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