## A Linear Time Approximation Algorithm for Weighted Matchings in Graphs (2003)

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Citations: | 19 - 3 self |

### BibTeX

@MISC{Drake03alinear,

author = {Doratha E. Drake and Stefan Hougardy},

title = {A Linear Time Approximation Algorithm for Weighted Matchings in Graphs },

year = {2003}

}

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### Abstract

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm+n 2 log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3

### Citations

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(Show Context)
Citation Context ...computed by starting with an empty matching and extending it in each step by an arbitrary edge in such a way that it remains a matching. Several variants of this simple algorithm are used in practice =-=[14]-=-. The advantage of maximal matching algorithms is that they have linear running time. The major disadvantage of these algorithms is that they have a performance ratio of 0, i.e., the solutions returne... |

618 |
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Citation Context ...l edge weights are integers in the range [1 . . . N] Gabow and Tarjan [13] presented an algorithm with running time O( � n · α(m, n) log n · m log(Nn)), where α is the inverse of Ackermann’s function =-=[24]-=-. In the case that all edge weights are the same, the fastest known algorithm has a running time of O( √ nm) and is due to Micali and Vazirani [21, 25]. For planar graphs Lipton and Tarjan [17] have s... |

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Citation Context ...blem was given by Edmonds [7] in 1965. A straightforward implementation of this algorithm requires a running time of O(n2m), where n and m denote the number of vertices and edges in the graph. Lawler =-=[15]-=- and Gabow [9] improved the running time to O(n3 ). Galil, Micali, and Gabow [12] presented an implementation of Edmond’s algorithm with a running time of O(nm log n). This was improved by Gabow, Gali... |

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Citation Context ...stest known algorithm has a running time of O( √ nm) and is due to Micali and Vazirani [21, 25]. For planar graphs Lipton and Tarjan [17] have shown that with the help of the Planar Separator Theorem =-=[16]-=- the weighted matching problem can be solved in O(n 3/2 log n). Together with the research on improving the worst case running time of Edmond’s algorithm for the weighted matching problem there has be... |

329 |
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Citation Context ...fined as w(M) := w(e). The weighted matching problem is to find a matching in G that has maximum e∈M weight. The first polynomial time algorithm for the weighted matching problem was given by Edmonds =-=[7]-=- in 1965. A straightforward implementation of this algorithm requires a running time of O(n2m), where n and m denote the number of vertices and edges in the graph. Lawler [15] and Gabow [9] improved t... |

168 |
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Citation Context ...nction [24]. In the case that all edge weights are the same, the fastest known algorithm has a running time of O( √ nm) and is due to Micali and Vazirani [21, 25]. For planar graphs Lipton and Tarjan =-=[17]-=- have shown that with the help of the Planar Separator Theorem [16] the weighted matching problem can be solved in O(n 3/2 log n). Together with the research on improving the worst case running time o... |

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Citation Context ... by Gabow, Galil, and Spencer [11] to a running time of O(nm log log log n + n2 log n). The fastest known algorithm to date for solving the weighted matching problem in general graphs is due to Gabow =-=[10]-=- and has a running time of O(nm + n2 log n). ⋆ supported by DFG research grant 296/6-3, supported in part by DFG Research Center Mathematics for key technologies, a preliminary version appeared in APP... |

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Citation Context ... in APPROX03 [6]sIn some special cases faster algorithms for the weighted matching problem are known. Under the assumption that all edge weights are integers in the range [1 . . . N] Gabow and Tarjan =-=[13]-=- presented an algorithm with running time O( � n · α(m, n) log n · m log(Nn)), where α is the inverse of Ackermann’s function [24]. In the case that all edge weights are the same, the fastest known al... |

89 | Computing minimum-weight perfect matchings
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Citation Context ...ures but also need additional new ideas to lower the running time in practice. During the last 35 years many different implementations of Edmond’s weighted matching algorithm have been presented. See =-=[3]-=- for a good survey on these. Currently the fastest implementations of Edmond’s algorithm are due to Cook and Rohe [3] and to Mehlhorn and Schäfer [20]. Many real world problems require graphs of such ... |

77 | An efficient implementation of Edmonds algorithm for maximum matching on graphs
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Citation Context ...by Edmonds [7] in 1965. A straightforward implementation of this algorithm requires a running time of O(n2m), where n and m denote the number of vertices and edges in the graph. Lawler [15] and Gabow =-=[9]-=- improved the running time to O(n3 ). Galil, Micali, and Gabow [12] presented an implementation of Edmond’s algorithm with a running time of O(nm log n). This was improved by Gabow, Galil, and Spencer... |

53 |
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Citation Context ...larly to the maximal matching algorithm but chooses in each step not an arbitrary but the heaviest edge currently available. It is easy to see that the greedy algorithm has a performance ratio of 1 2 =-=[1]-=-. The running time of this algorithm is O(m log n) as it requires sorting the edges of the graph by decreasing weight. Preis [22] was the first who was able to combine the advantages of the greedy alg... |

49 |
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Citation Context ...Then we have w(Mi+1) ≥ wi+1 · w(Mopt) where wi+1 is defined by the following recurrence wi+1 = wi + β − 1 2β � � 2 − wi , and w0 = 3β 1 2 . By solving this linear recurrence equation (see for example =-=[23]-=-) we get � 2 w(Mk) ≥ 3β + � � � � � k 1 2 1 1 − + · w(Mopt) . 2 3β 2 2β This shows that for any fixed β and sufficiently large k (depending only on β) there exists a linear time algorithm which finds ... |

38 | Linear time 1/2-approximation algorithm for maximum weighted matching in general graphs,” ser
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Citation Context ...is easy to see that the greedy algorithm has a performance ratio of 1 2 [1]. The running time of this algorithm is O(m log n) as it requires sorting the edges of the graph by decreasing weight. Preis =-=[22]-=- was the first who was able to combine the advantages of the greedy algorithm and the maximal matching algorithm in one algorithm. In 1999 he presented a linear time approximation algorithm for the we... |

36 |
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Citation Context ...improved the running time to O(n3 ). Galil, Micali, and Gabow [12] presented an implementation of Edmond’s algorithm with a running time of O(nm log n). This was improved by Gabow, Galil, and Spencer =-=[11]-=- to a running time of O(nm log log log n + n2 log n). The fastest known algorithm to date for solving the weighted matching problem in general graphs is due to Gabow [10] and has a running time of O(n... |

34 | A simple approximation algorithm for the weighted matching problem, Information Processing letters 85
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Citation Context ... difficult to prove that finding a locally heaviest edge in each step can be done in such a way that the total running time remains linear. By using a completely different approach Drake and Hougardy =-=[4]-=- obtained another linear time approximation algorithm for the weighted matching problem with a performance ratio of 1 2 . The main idea of their algorithm is to grow in a greedy way two matchings simu... |

29 | Quality matching and local improvement for multilevel graph-partitioning
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Citation Context ...e size that the running time of the fastest available weighted matching algorithm is too costly. Examples of such problems are the refinement of FEM nets [18], the partitioning problem in VLSI-Design =-=[19]-=-, and the gossiping problem in telecommunications [2]. There also exist applications were the weighted matching problem has to be solved extremely often on only moderately large graphs. An example of ... |

28 |
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Citation Context ...lgorithm requires a running time of O(n2m), where n and m denote the number of vertices and edges in the graph. Lawler [15] and Gabow [9] improved the running time to O(n3 ). Galil, Micali, and Gabow =-=[12]-=- presented an implementation of Edmond’s algorithm with a running time of O(nm log n). This was improved by Gabow, Galil, and Spencer [11] to a running time of O(nm log log log n + n2 log n). The fast... |

13 |
An O( |V |·|E|) Algorithm for Finding Maximum Matching
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Citation Context ...Nn)), where α is the inverse of Ackermann’s function [24]. In the case that all edge weights are the same, the fastest known algorithm has a running time of O( √ nm) and is due to Micali and Vazirani =-=[21, 25]-=-. For planar graphs Lipton and Tarjan [17] have shown that with the help of the Planar Separator Theorem [16] the weighted matching problem can be solved in O(n 3/2 log n). Together with the research ... |

9 | Complexity and Modeling Aspects of Mesh Refinement into Quadrilater
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Citation Context ...eal world problems require graphs of such large size that the running time of the fastest available weighted matching algorithm is too costly. Examples of such problems are the refinement of FEM nets =-=[18]-=-, the partitioning problem in VLSI-Design [19], and the gossiping problem in telecommunications [2]. There also exist applications were the weighted matching problem has to be solved extremely often o... |

7 | Accelerating screening of 3d protein data with a graph theoretical approach
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- 2003
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Citation Context ...o be solved extremely often on only moderately large graphs. An example of such an application is the virtual screening of protein databases containing the three dimensional structure of the proteins =-=[8]-=-. The graphs appearing in such applications have only about 10,000 edges, but the weighted matching problem has to be solved more than 100,000,000 times for a complete database scan. Therefore, there ... |

6 |
G.: Implementation of O(nmlogn) weighted matchings in general graphs: the power of data structures
- Mehlhorn, Schäfer
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(Show Context)
Citation Context ...ighted matching algorithm have been presented. See [3] for a good survey on these. Currently the fastest implementations of Edmond’s algorithm are due to Cook and Rohe [3] and to Mehlhorn and Schäfer =-=[20]-=-. Many real world problems require graphs of such large size that the running time of the fastest available weighted matching algorithm is too costly. Examples of such problems are the refinement of F... |

4 | A Powerful Heuristic for Telephone Gossiping
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- 2000
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Citation Context ...weighted matching algorithm is too costly. Examples of such problems are the refinement of FEM nets [18], the partitioning problem in VLSI-Design [19], and the gossiping problem in telecommunications =-=[2]-=-. There also exist applications were the weighted matching problem has to be solved extremely often on only moderately large graphs. An example of such an application is the virtual screening of prote... |

3 |
Linear Time Local Improvements for Weighted
- Drake, Hougardy
- 2003
(Show Context)
Citation Context ... main idea of their algorithm is to grow in a greedy way two matchings simultaneously and return the heavier of both as the result. Their algorithm and its analysis are simpler than that of Preis. In =-=[5]-=- the idea of local improvements is used as a postprocessing step to enhance the performance of approximation algorithms for the weighted matching problem in practice. This postprocessing step requires... |

1 |
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Citation Context ...Nn)), where α is the inverse of Ackermann’s function [24]. In the case that all edge weights are the same, the fastest known algorithm has a running time of O( √ nm) and is due to Micali and Vazirani =-=[21, 25]-=-. For planar graphs Lipton and Tarjan [17] have shown that with the help of the Planar Separator Theorem [16] the weighted matching problem can be solved in O(n 3/2 log n). Together with the research ... |

1 | Implementation of O(nmlogn) WeightedMatchingsinGeneralGraphs:The Power of Data Structures - Mehlhorn, Schäfer |

1 | An O( √ VE)AlgorithmforFindingMaximumMatchinginGeneral Graphs - Micali, Vazirani - 1980 |