The fundamental Sparsest Cut problem takes as input a graph G together with edge capacities and demands, and seeks a cut that minimizes the ratio between the capacities and demands across the cuts. For n-vertex graphs G of treewidth k, Chlamtáč, Krauthgamer, and Raghavendra (APPROX 2010) presented an algorithm that yields a factor-22k approximation in time 2O(k) · nO(1). Later, Gupta, Talwar and Witmer (STOC 2013) showed how to obtain a 2-approximation algorithm with a blown-up run time of nO(k). An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-2 approximation in time 2O(k) · nO(1). In this paper, we make significant progress towards this goal, via the following results: (i) A factor-O(k2) approximation that runs in time 2O(k) · nO(1), directly improving the work of Chlamtáč et al. while keeping the run time single-exponential in k. (ii) For any ε ∈ (0, 1], a factor-O(1/ε2) approximation whose run time is 2O(k1+ε/ε)⋅nO(1), implying a constant-factor approximation whose run time is nearly single-exponential in k and a factor-O(log 2k) approximation in time kO(k) · nO(1). Key to these results is a new measure of a tree decomposition that we call combinatorial diameter, which may be of independent interest.

A 2-walk of a graph is a walk visiting every vertex at least once and at most twice. By generalizing decompositions of Tutte and Thomassen, Gao, Richter and Yu proved that every 3-connected planar graph contains a closed 2-walk such that all vertices visited twice are contained in 3-separators. This seminal result generalizes Tutte's theorem that every 4-connected planar graph is Hamiltonian as well as Barnette's theorem that every 3-connected planar graph has a spanning tree with maximum degree at most 3. The algorithmic challenge of finding such a closed 2-walk is to overcome big overlapping subgraphs in the decomposition, which are also inherent in Tutte's and Thomassen's decompositions. We solve this problem by extending the decomposition of Gao, Richter and Yu in such a way that all pieces, into which the graph is decomposed, are edge-disjoint. This implies the first polynomial-time algorithm that computes the closed 2-walk mentioned above. Its running time is O(n³).