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

2021 ◽  
Vol 182 (3) ◽  
pp. 257-283
Author(s):  
Viet Dung Nguyen ◽  
Ba Thai Pham ◽  
Phan Thuan Do
Keyword(s):  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Matching Problem ◽  
Induced Matching ◽  
Graph Classes ◽  
Permutation Models ◽  
Maximum Induced Matching ◽  
Sweep Line ◽  
Trapezoid Graphs ◽  
Better Than

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.

scholarly journals On the induced matching problem in hamiltonian bipartite graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinglei Song
Keyword(s):  
Bipartite Graph ◽  
Parameterized Complexity ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Induced Matching ◽  
Subexponential Time ◽  
Maximum Induced Matching

Abstract 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.

scholarly journals Latest Algorithms on Particular Graph Classes

2020 ◽  
pp. 21-35
Author(s):  
Phan Thuan DO ◽  
Ba Thai PHAM ◽  
Viet Cuong THAN
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Maximum Clique ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Np Hard ◽  
Induced Matching ◽  
Graph Classes ◽  
Maximum Induced Matching ◽  
Circular Arc Graphs

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.

Improved Computation of Optimal Rectilinear Steiner Minimal Trees

1997 ◽  
Vol 07 (05) ◽  
pp. 457-472 ◽  
Author(s):  
Joseph L. Ganley ◽  
James P. Cohoon
Keyword(s):  
Dynamic Programming ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
Steiner Minimal Trees ◽  
Previous Algorithm ◽  
Time Bounds ◽  
New Algorithms ◽  
Better Than

This paper presents two new algorithms for computing optimal rectilinear Steiner minimal trees. The first algorithm is a simple, easily implemented dynamic programming algorithm that computes an optimal rectilinear Steiner minimal tree on n points in O(n3n) time. The second algorithm improves upon the first using the concept of full-set screening and runs in at most O(n22.62n) time. The analysis of the second algorithm includes a proof that there are at most O(n 1.62n) full sets on n terminals. For instances small enough to practically solve, these time bounds are provably better than the best known bounds of any previous algorithm. Experimental evidence is presented that demonstrates that the algorithms are fast in practice as well. The paper also includes a brief survey of previous algorithms for computing optimal rectilinear Steiner minimal trees.

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