Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter,ACM Transactions on Algorithms

当前位置： X-MOL 学术 › ACM Trans. Algorithms › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter
ACM Transactions on Algorithms ( IF 1.3 ) Pub Date : 2023-11-14 , DOI: 10.1145/3632623
Parinya Chalermsook ₁ , Matthias Kaul ₂ , Matthias Mnich ₂ , Joachim Spoerhase ₃ , Sumedha Uniyal ₁ , Daniel Vaz ₄

Affiliation

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 2010) presented an algorithm that yields a factor-\(2^{2^k} \) approximation in time 2^O(k) · n^O(1). 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) · n^O(1).

In this paper, we make significant progress towards this goal, via the following results:

(i)

	A factor-O(k²) approximation that runs in time 2^O(k) · n^O(1), directly improving the work of Chlamtáč et al. while keeping the run time single-exponential in k.
(ii)	For any ε ∈ (0, 1], a factor-O(1/ε²) approximation whose run time is \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)} \), implying a constant-factor approximation whose run time is nearly single-exponential in k and a factor-O(log ²k) approximation in time k^O(k) · 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.

中文翻译：

通过组合直径近似低树宽图中的最稀疏割

基本的稀疏割问题将图G以及边容量和需求作为输入，并寻求一种割，使割中的容量和需求之间的比率最小化。对于树宽为 k的n顶点图 G ，Chlamtáč、Krauthgamer 和 Raghavendra (APPROX 2010) 提出了一种算法，可在 2 ^O^{( k}⁾ · n ^{O时间内生成因子-\(2^{2^}^k } \) 近似值⁽¹⁾ . 后来，Gupta、Talwar 和 Witmer (STOC 2013) 展示了如何获得运行时间为n ^O⁽^k⁾的 2 近似算法。一个有趣的悬而未决的问题是，是否可以同时实现上述结果中的最佳结果，即时间 2 ^O⁽^k⁾ · n ^O⁽¹⁾的因子 2 近似。

在本文中，我们通过以下结果在实现这一目标方面取得了重大进展：

（我）

	^{在 2 O}⁽^k⁾ · n ^O⁽¹⁾时间内运行的因子O ( k ² ) 近似，直接改进了 Chlamtáč 等人的工作。同时保持k中的运行时间为单指数。
(二)	对于任何 ε ε (0, 1]，一个因子O (1/ε ² ) 近似，其运行时间为 \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{ O(1)} \)，意味着运行时间接近k的单指数常数因子近似，以及时间k ^O⁽^k⁾ · n ^O^{(1) 的}因子O (log ²k ) 近似。