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 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 Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness results: If the Non-Uniform Spars-est 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.