Efficient algorithms for maximum induced matching problem in permutation and trapezoid graphs

TL;DR

This paper presents improved algorithms for the maximum induced matching problem in permutation and trapezoid graphs, achieving linear and near-linear time complexities, significantly outperforming previous methods.

Contribution

It introduces the first linear-time algorithm for permutation graphs and an improved near-linear algorithm for trapezoid graphs using dynamic programming and disjoint-set data structures.

Findings

Linear-time algorithm for permutation graphs.

Near-linear time algorithm for trapezoid graphs.

Significant complexity reduction over previous methods.

Abstract

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.