Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter

The fundamental sparsest cut problem takes as input a graph \(G\) together with the edge costs and demands, and seeks a cut that minimizes the ratio between the costs and demands across the cuts. For \(n\)-node graphs~\(G\) of treewidth~\(k\), \chlamtac, Krauthgamer, and Raghavendra (APPROX 2010) presented an algorithm that yields a factor-\(2^{2^k}\) approximation in time \(2^{O(k)} \cdot \operatorname{poly}(n)\). 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 \(2^{O(k)} \cdot \operatorname{poly}(n)\). 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 \(2^{O(k)} \cdot \operatorname{poly}(n)\), directly improving the work of Chlamtáč et al. while keeping the run time single-exponential in \(k\). (ii) For any \(\varepsilon>0\), a factor-\(O(1/\varepsilon^2)\) approximation whose run time is \(2^{O(k^{1+\varepsilon}/\varepsilon)} \cdot \operatorname{poly}(n)\), implying a constant-factor approximation whose run time is nearly single-exponential in \(k\) and a factor-\(O(\log^2 k)\) approximation in time \(k^{O(k)} \cdot \operatorname{poly}(n)\). Key to these results is a new measure of a tree decomposition that we call combinatorial diameter, which may be of independent interest.