@inproceedings{6ecdedf9f2834ceeb16b813c0a3a63da,
title = "Sparsest cut on bounded treewidth graphs: Algorithms and hardness results",
abstract = "We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 k-approximation in time poly(n) 2O(k) due to Chlamt{\'a}̌c et al. [18]. To complement this algorithm, we show the following hardness results: If the Non-Uniform Sparsest Cut problem has a ρ-approximation for series-parallel graphs (where ρ ≥ 1), then the MAXCUT problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard to approximate better than 17/16-ε for ε> 0; assuming the Unique Games Conjecture the hardness becomes 1/αGW - ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 - ε 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.",
keywords = "APX-hardness, Graph separators, Sherali-adams, Sparsest cut, Treewidth, Unique games",
author = "Anupam Gupta and Kunal Talwar and David Witmer",
year = "2013",
doi = "10.1145/2488608.2488644",
language = "English (US)",
isbn = "9781450320290",
series = "Proceedings of the Annual ACM Symposium on Theory of Computing",
pages = "281--290",
booktitle = "STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing",
note = "45th Annual ACM Symposium on Theory of Computing, STOC 2013 ; Conference date: 01-06-2013 Through 04-06-2013",
}