Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter

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- \(2^{2^k} \) approximation in time 2 O ( k ) · n O (1) . Later, Gupta, Talwar and Witmer (STOC 2013) showed how to obtain a 2-approximation algorithm with a blown-up run time of n O ( 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 2 O ( k ) · n O (1) . In this paper, we make significant progress towards this goal, via the following results: (i) A factor- O ( k 2 ) approximation that runs in time 2 O ( k ) · n O (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/ε 2 ) approximation whose run time is \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)} \) , implying a constant-factor approximation whose run time is nearly single-exponential in k and a factor- O (log 2 k ) approximation in time k O ( k ) · n O (1) . Key to these results is a new measure of a tree decomposition that we call combinatorial diameter , which may be of independent interest.