Parinya Chalermsook
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:
Key to these results is a new measure of a tree decomposition that we call combinatorial diameter, which may be of independent interest.
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Index Terms
- Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter
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