Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter

Abstract: 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-$2<sup>{2<sup>k}$ 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 $n<sup>{O(k)}$. 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-$O(k<sup>2)$ approximation that runs in time $2^{O(k)} \cdot \operatorname{poly}(n)$, directly improving the work of Chlamt\'a\v{c} et al. while keeping the run time single-exponential in $k$. (ii) For any $\varepsilon&gt;0$, a factor-$O(1/\varepsilon<sup>2)$ 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-$O(\log² k)$ 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.