Efficient Algorithms for Maximum Induced Matching Problem in Permutation and Trapezoid Graphs

We first design an 𝒪(n2) 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 𝒪(m+n). Consequently, we extend this result to give an 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 𝒪(n2) time, respectively. Our results are far better than the best known 𝒪(mn) algorithm for the maximum induced matching problem in both graph classes, which was proposed by Habib et al.

On the induced matching problem in hamiltonian bipartite graphs

In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

Latest Algorithms on Particular Graph Classes

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

Wu et al. (2014) showed that under the small set expansion hypothesis (SSEH) there is no polynomial time approximation algorithm with any constant approximation factor for several graph width parameters, including tree-width, path-width, and cut-width (Wu et al. 2014). In this paper, we extend this line of research by exploring other graph width parameters: We obtain similar approximation hardness results under the SSEH for rank-width and maximum induced matching-width, while at the same time we show the approximation hardness of carving-width, clique-width, NLC-width, and boolean-width. We also give a simpler proof of the approximation hardness of tree-width, path-width, and cut-widththan that of Wu et al.