Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results

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(2k)-approximation in time poly(n) 2(O(k)) due to Chlamtac et al. [18]. 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 >= 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.