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 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 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.

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

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.

A New Algorithm for Subset Matching Problem Based on Set-String Transformation

2009 ◽  
pp. 607-615
Author(s):  
Yangjun Chen
Keyword(s):  
Programming Languages ◽  
Linear Time ◽  
String Matching ◽  
Special Problem ◽  
Computer Engineering ◽  
Data Types ◽  
Matching Problem ◽  
Matching Problems ◽  
Abstract Data ◽  
Geometric Pattern Matching

In computer engineering, a number of programming tasks involve a special problem, the so-called tree matching problem (Cole & Hariharan, 1997), as a crucial step, such as the design of interpreters for nonprocedural programming languages, automatic implementation of abstract data types, code optimization in compilers, symbolic computation, context searching in structure editors and automatic theorem proving. Recently, it has been shown that this problem can be transformed in linear time to another problem, the so called subset matching problem (Cole & Hariharan, 2002, 2003), which is to find all occurrences of a pattern string p of length m in a text string t of length n, where each pattern and text position is a set of characters drawn from some alphabet S. The pattern is said to occur at text position i if the set p[j] is a subset of the set t[i + j - 1], for all j (1 = j = m). This is a generalization of the ordinary string matching and is of interest since an efficient algorithm for this problem implies an efficient solution to the tree matching problem. In addition, as shown in (Indyk, 1997), this problem can also be used to solve general string matching and counting matching (Muthukrishan, 1997; Muthukrishan & Palem, 1994), and enables us to design efficient algorithms for several geometric pattern matching problems. In this article, we propose a new algorithm on this issue, which needs only O(n + m) time in the case that the size of S is small and O(n + m·n0.5) time on average in general cases.

Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

2013 ◽  
Vol 22 (5) ◽  
pp. 783-799 ◽  
Author(s):  
GUILLEM PERARNAU ◽  
ORIOL SERRA
Keyword(s):  
Bipartite Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Complete Bipartite Graphs ◽  
Edge Colourings ◽  
Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

Finite locally-primitive complete bipartite graphs

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Wenwen Fan ◽  
Cai Heng Li ◽  
Jiangmin Pan
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Complete Bipartite Graphs

Abstract.We characterize groups which act locally-primitively on a complete bipartite graph. The result particularly determines certain interesting factorizations of groups.

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