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

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.

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

10.21203/rs.3.rs-19322/v1 ◽

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.

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On the induced matching problem in hamiltonian bipartite graphs

Georgian Mathematical Journal ◽

10.1515/gmj-2021-2090 ◽

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.

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Augmenting trail theorem for the maximum 1-2 matching problem

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830917500562 ◽

2017 ◽

Vol 09 (04) ◽

pp. 1750056 ◽

Cited By ~ 1

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.

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ACYCLIC MATCHINGS IN SUBCLASSES OF BIPARTITE GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500504 ◽

2012 ◽

Vol 04 (04) ◽

pp. 1250050 ◽

Cited By ~ 4

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.

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A Note on $\mathtt{V}$-free 2-matchings

The Electronic Journal of Combinatorics ◽

10.37236/5258 ◽

2016 ◽

Vol 23 (4) ◽

Author(s):

Kristóf Bérczi ◽

Attila Bernáth ◽

Máté Vizer

Keyword(s):

Bipartite Graph ◽

Packing Problem ◽

Bipartite Graphs ◽

Matching Problem ◽

Hypergraph Matching ◽

Path Packing ◽

Np Complete

Motivated by a conjecture of Liang, we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of the hypergraph matching problem. We close the paper by showing that it is NP-complete to decide whether one of the color classes of a bipartite graph can be covered by a $\mathtt{V}$-free $2$-matching.

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