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• W2951567887 abstract "We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time $n^{O(k)}$, where $k$ is the treewidth of the graph. This improves on the previous $2^{2^k}$-approximation in time $poly(n) 2^{O(k)}$ due to Chlamt'av{c} et al. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a $rho$-approximation for series-parallel graphs (where $rho geq 1$), then the Max Cut problem has an algorithm with approximation factor arbitrarily close to $1/rho$. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than $17/16 - epsilon$ for $epsilon > 0$; assuming the Unique Games Conjecture the hardness becomes $1/alpha_{GW} - epsilon$. For graphs with large (but constant) treewidth, we show a hardness result of $2 - epsilon$ assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for treewidth-2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of Sherali-Adams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation." @default.
• W2951567887 created "2019-06-27" @default.
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• W2951567887 date "2013-05-06" @default.
• W2951567887 modified "2023-09-27" @default.
• W2951567887 title "Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results" @default.
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