Induced Matching in Some Subclasses of Bipartite Graphs

Pandey, Arti; Panda, B. S.; Dane, Piyush; Kashyap, Manav

doi:10.1007/978-3-319-53007-9_27

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…All of our algorithms achieve optimal running time for the respective problem and model. Our results for finding a maximum-cardinality induced matching also improve the running times of the algorithms of Pandey et al [21] for the circular-convex and triad-convex case, as they use the convex case as a building block.…”

Section: Related Work and Motivationsupporting

confidence: 58%

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Klemz

Rote

2022

Algorithmica

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A bipartite graph $$G=(U,V,E)$$ G = ( U , V , E ) is convex if the vertices in V can be linearly ordered such that for each vertex $$u\in U$$ u ∈ U , the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in $$O(n+m)$$ O ( n + m ) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex $$u\in U$$ u ∈ U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in $$O(n+m)$$ O ( n + m ) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of $$O(n^2)$$ O ( n 2 ) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. 10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.

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Section: Related Work and Motivationsupporting

confidence: 58%

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Klemz

Rote

2022

Algorithmica

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show abstract

“…All of our algorithms achieve optimal running time for the respective problem and model. Our results for finding a maximum-cardinality induced matching also improve the running times of the algorithms of Pandey et al [19] for the circular-convex and triad-convex case, as they use the convex case as a building block.…”

Section: Introductionsupporting

confidence: 58%

“…Chang [6] computes maximum-cardinality induced matchings and minimum chain covers in O(n + m) time in bipartite permutation graphs, which form a proper subclass of convex bipartite graphs. Recently, Pandey, Panda, Dane and Kashyap [19] gave polynomial algorithms for finding a maximum-cardinality induced matching in circular-convex and triad-convex bipartite graphs. These graph classes generalize convex bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Klemz¹,

Rote²

2017

Preprint

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A bipartite graph G = (U, V, E) is convex if the vertices in V can be linearly ordered such that for each vertex u ∈ U , the neighbors of u are consecutive in the ordering of V . An induced matching H of G is a matching such that no edge of E connects endpoints of two different edges of H.We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n + m) time.An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u ∈ U the first and last neighbor in the ordering of V . Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time.In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in O(n + m) time.All of our algorithms achieve optimal running time for the respective problem and model. Previous algorithms considered only the unweighted case, and the best algorithm for computing a maximum-cardinality induced matching or a minimum chain cover in a convex bipartite graph had a running time of O(n 2 ).

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“…Besides, it is known for polynomial-time maximum induced matching on special graph classes such as co-comparability graphs (including circular-arc graphs [10], interval graphs [11], etc. ), circular-convex bipartite and triad-convex bipartite graphs [12], AT-free graphs [13] and hexagonal graphs [14]. In chordal graphs, finding a MIM can be done in linear time [15].…”

Section: Introductionmentioning

confidence: 99%

“…A permutation graph G (a) and one of its corresponding permutation representation (b), which has π =(5,7,2,1,4,8,11,10,3,6,12,9). A trapezoid model of the trapezoid graph L(G) 2 , partially shown in (c), can be constructed from the permutation representation of G. A maximum induced matching for G is M = {(1, 5),(6,8),(9,12)}, also seen as a maximum independent set of L(G)2 . A trapezoid graph G (a) and one of its corresponding trapezoid representations (b).…”

mentioning

confidence: 99%

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

Nguyen,

Pham,

2021

Preprint

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We first design an 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 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(n 2 ) 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.

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Induced Matching in Some Subclasses of Bipartite Graphs

Cited by 6 publications

References 18 publications

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

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

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