The Cardinality Matching Problem in Bipartite Graphs

1988 ◽  
pp. 121-133
Author(s):  
Ulrich Derigs
Keyword(s):  
Bipartite Graphs ◽  
Matching Problem

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.

Augmenting trail theorem for the maximum 1-2 matching problem

2017 ◽  
Vol 09 (04) ◽  
pp. 1750056 ◽  
Author(s):  
Hiroki Izumi ◽  
Sennosuke Watanabe ◽  
Yoshihide Watanabe
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Matching Problems ◽  
Augmenting Path

We consider the maximum 1-2 matching problem in bipartite graphs. The notion of the augmenting trail for the 1-2 matching problem, which is the extension of the notion of the augmenting path for the 1-1 matching problem is introduced. The main purpose of the present paper is to prove “the augmenting trail theorem” for the 1-2 matching problem in the bipartite graph, which is an analogue of the augmenting path theorem by Bergé for the usual 1-1 matching problems.

scholarly journals An O(n^3) time algorithm for the maximum weight b-matching problem on bipartite graphs

2020 ◽  
Author(s):  
Fatemeh Rajabi-Alni ◽  
Alireza Bagheri ◽  
Behrouz Minaei-Bidgoli
Keyword(s):  
Computational Biology ◽  
Bipartite Graph ◽  
Association Studies ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Maximum Weight ◽  
Positive Real ◽  
Matching Problem ◽  
Genome Wide ◽  
Integer Weight

Abstract Background: A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology for example in the contexts of genome-wide and sequencing association studies. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. Results: We use bipartite graphs to model relationships between pairs of objects. Given an undirected bipartite graph G = (A [ B;E), the b-matching of G matches each vertex v in A (resp. B) to at least 1 and at most b(v) vertices in B (resp. A), where b(v) denotes the capacity of v. We propose the rst O(n3) time algorithm for nding the maximum weight b-matching of G, where jAj + jBj = O(n). Conclusions: The b-matching has been studied widely for the bipartite graphs with integer weight edges. But our algorithm is the rst algorithm for the maximum (respectively minimum) b-matching problem with non positive real (respectively non negative real) edge weights.

scholarly journals An O(n3) time algorithm for the maximum weight b-matching problem on bipartite graphs

2020 ◽  
Author(s):  
Fatemeh Rajabi-Alni ◽  
Alireza Bagheri ◽  
Behrouz Minaei-Bidgoli
Keyword(s):  
Computational Biology ◽  
Bipartite Graph ◽  
Association Studies ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Maximum Weight ◽  
Positive Real ◽  
Matching Problem ◽  
Genome Wide ◽  
Integer Weight

Abstract Background: A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology for example in the contexts of genome-wide and sequencing association studies. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. Results: We use bipartite graphs to model relationships between pairs of objects. Given an undirected bipartite graph G = (A∪B,E), the b-matching of G matches each vertex v in A (resp. B) to at least 1 and at most b(v) vertices in B (resp. A), where b(v) denotes the capacity of v. We propose the first O(n3) time algorithm for finding the maximum weight b-matching of G, where |A|+|B| = O(n). Conclusions: The b-matching has been studied widely for the bipartite graphs with integer weight edges. But our algorithm is the first algorithm for the maximum (respectively minimum) b-matching problem with non positive real (respectively non negative real) edge weights.

The maximum 1-2 matching problem and two kinds of its variants

2019 ◽  
Vol 11 (05) ◽  
pp. 1950050
Author(s):  
Hiroki Izumi ◽  
Yuki Nishida ◽  
Sennosuke Watanabe ◽  
Yoshihide Watanabe
Keyword(s):  
Bipartite Graphs ◽  
The Other ◽  
Undirected Graphs ◽  
Matching Problem

We present an algorithm for solving the maximum 1-2 matching problem in bipartite graphs by reducing it to solving the 1-1 matching in general undirected graphs. Further, we propose two variants of the maximum 1-2 matching problem: one is the maximum 1-2 matching problem of the different kind in divided bipartite graphs and the other is the maximum 1-2 matching problem of the same kind in divided bipartite graphs. The former one is shown to be solved by a similar algorithm to that of the maximum 1-2 matching problem by reducing it to solve the maximum 1-1 matching problem in bipartite graphs. For the latter one, we can only present the augmenting subgraph theorem, which is an extension of our augmenting trail theorem for the maximum 1-2 matching problem.

ACYCLIC MATCHINGS IN SUBCLASSES OF BIPARTITE GRAPHS

2012 ◽  
Vol 04 (04) ◽  
pp. 1250050 ◽  
Author(s):  
B. S. PANDA ◽  
D. PRADHAN
Keyword(s):  
Time Algorithm ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Chain Graph ◽  
Matching Problem ◽  
Permutation Graph ◽  
Complexity Result ◽  
Np Complete ◽  
A Chain ◽  
Acyclic Matching

A set M ⊆ E is called an acyclic matching of a graph G = (V, E) if no two edges in M are adjacent and the subgraph induced by the set of end vertices of the edges of M is acyclic. Given a positive integer k and a graph G = (V, E), the acyclic matching problem is to decide whether G has an acyclic matching of cardinality at least k. Goddard et al. (Discrete Math.293(1–3) (2005) 129–138) introduced the concept of the acyclic matching problem and proved that the acyclic matching problem is NP-complete for general graphs. In this paper, we propose an O(n + m) time algorithm to find a maximum cardinality acyclic matching in a chain graph having n vertices and m edges and obtain an expression for the number of maximum cardinality acyclic matchings in a chain graph. We also propose a dynamic programming-based O(n + m) time algorithm to find a maximum cardinality acyclic matching in a bipartite permutation graph having n vertices and m edges. Finally, we strengthen the complexity result of the acyclic matching problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs.

scholarly journals Maximum semi-matching problem in bipartite graphs

2013 ◽  
Vol 33 (3) ◽  
pp. 559 ◽  
Author(s):  
Ján Katrenič ◽  
Gabriel Semanišin
Keyword(s):  
Bipartite Graphs ◽  
Matching Problem
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