Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter
Parinya Chalermsook, Matthias Kaul, Matthias Mnich, Joachim Spoerhase, Sumedha Uniyal, Daniel Vaz
TL;DR
This work studies sparsest cut on graphs with small treewidth by introducing combinatorial diameter as a structural parameter of tree decompositions. The authors prove that CKR-type rounding yields an approximation factor bounded by $\min\{O(\Delta(\mathcal{T})^2), 2^{2^{w(\mathcal{T})}}\}$, and then construct three decompositions achieving diverse $w(\mathcal{T})$ and $\Delta(\mathcal{T})$ tradeoffs. Consequently, they obtain (i) a factor $O(k^2)$ in time $2^{O(k)}\cdot\mathrm{poly}(n)$, (ii) a constant-factor approximation in time $2^{O(k^2)}\cdot\mathrm{poly}(n)$, and (iii) an approximation scheme with nearly single-exponential run time $\exp(O(k^{1+\varepsilon}/\varepsilon))$ and factor $O(1/\varepsilon^2)$, plus a variant achieving $O(\log^2 k)$ in time $k^{O(k)}\cdot\mathrm{poly}(n)$. The combinatorial diameter measure could be of independent interest beyond sparsest cut.
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-$2^{2^k}$ 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.