Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter

Parinya Chalermsook, Matthias Kaul, Matthias Mnich, Joachim Spoerhase, Sumedha Uniyal, Daniel Vaz

Abstract: 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’10) presented an algorithm that yields a factor-\(2^{2^k}\) approximation in time \(2^{O(k)} \cdot n^{O(1)}\). Later, Gupta, Talwar, and Witmer (STOC’13) showed how to obtain a 2-approximation algorithm with a blown-up runtime 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 \(2^{O(k)} \cdot n^{O(1)}\).In this article, we make significant progress towards this goal via the following results: (i)A factor-\(O(k^2)\) approximation that runs in time \(2^{O(k)} \cdot n^{O(1)}\), directly improving the work of Chlamtáč et al. while keeping the runtime single-exponential in k.(ii)For any \(\varepsilon \in (0,1]\), a factor-\(O(1/\varepsilon ^2)\) approximation whose runtime is \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)}\), implying a constant-factor approximation whose runtime is nearly single-exponential in k and a factor-\(O(\log ^2 k)\) approximation in time \(k^{O(k)} \cdot n^{O(1)}\).Key to these results is a new measure of a tree decomposition that we call combinatorial diameter, which may be of independent interest.