Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter
Authors: 
Parinya Chalermsook, Matthias Kaul, Matthias Mnich, Joachim Spoerhase, Sumedha Uniyal, Daniel VazAuthors Info & Claims
Parinya Chalermsook, Matthias Kaul, Matthias Mnich, Joachim Spoerhase, Sumedha Uniyal, Daniel VazAuthors Info & ClaimsArticle No.: 6, Pages 1 - 20
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.
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- Approximating Sparsest Cut in Low-treewidth Graphs via Combinatorial Diameter
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Publication History
Published: 22 January 2024
Online AM: 14 November 2023
Accepted: 30 October 2023
Revised: 09 October 2023
Received: 26 September 2021
Published in TALG Volume 20, Issue 1
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- European Research Council (ERC)
- European Union’s Horizon 2020
- Academy of Finland Research Fellowship
- French National Research Agency (ANR)
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