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Efficient algorithms for maximum induced matching problem in permutation and trapezoid graphs
arXiv - CS - Discrete Mathematics Pub Date : 2021-07-18 , DOI: arxiv-2107.08480
Viet Dung Nguyen, Ba Thai Pham, Phan Thuan Do

We first design an $\mathcal{O}(n^2)$ solution for finding a maximum induced matching in permutation graphs given their permutation models, based on a dynamic programming algorithm with the aid of the sweep line technique. With the support of the disjoint-set data structure, we improve the complexity to $\mathcal{O}(m + n)$. Consequently, we extend this result to give an $\mathcal{O}(m + n)$ algorithm for the same problem in trapezoid graphs. By combining our algorithms with the current best graph identification algorithms, we can solve the MIM problem in permutation and trapezoid graphs in linear and $\mathcal{O}(n^2)$ time, respectively. Our results are far better than the best known $\mathcal{O}(mn)$ algorithm for the maximum induced matching problem in both graph classes, which was proposed by Habib et al.

中文翻译:

排列和梯形图中最大诱导匹配问题的有效算法

我们首先设计了一个 $\mathcal{O}(n^2)$ 解决方案,用于在给定排列模型的情况下基于动态规划算法在扫描线技术的帮助下找到排列图中的最大诱导匹配。在不相交集数据结构的支持下,我们将复杂度提高到 $\mathcal{O}(m + n)$。因此,我们扩展了这个结果,为梯形图中的同一问题给出了一个 $\mathcal{O}(m + n)$ 算法。通过将我们的算法与当前最好的图识别算法相结合,我们可以分别在线性和 $\mathcal{O}(n^2)$ 时间内解决排列和梯形图中的 MIM 问题。我们的结果远远好于 Habib 等人提出的用于解决两个图类中最大诱导匹配问题的最著名的 $\mathcal{O}(mn)$ 算法。
更新日期:2021-07-20
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