New Iterative Algorithms for Weighted Matching

Matching is an important combinatorial problem with a number ofpractical applications. Even though there exist polynomial time solutionsto most matching problems, there are settings where these are too slow.This has led to the development of several fast approximation algorithmsthat in practice compute matchings very close to the optimal.The current paper introduces a new deterministic approximationalgorithm named G 3 , for weighted matching. The algorithm is based ontwo main ideas, the first is to compute heavy subgraphs of the originalgraph on which we can compute optimal matchings. The second idea isto repeatedly merge these matchings into new matchings of even higherweight than the original ones. Both of these steps are achieved usingdynamic programming in linear or close to linear time.We compare G 3 with the randomized algorithm GPA+ROMA whichis the best known algorithm for this problem. Experiments on alarge collection of graphs show that G 3 is substantially faster thanGPA+ROMA while computing matchings of comparable weight

A New Self-Stabilizing Maximal Matching Algorithm

The maximal matching problem has received considerable attention in the self-stabilizing community. Previous work has given different self-stabilizing algorithms that solves the problem for both the adversarial and fair distributed daemon, the sequential adversarial daemon, as well as the synchronous daemon. In the following we present a single self-stabilizing algorithm for this problem that unites all of these algorithms in that it stabilizes in the same number of moves as the previous best algorithms for the sequential adversarial, the distributed fair, and the synchronous daemon. In addition, the algorithm improves the previous best moves complexities for the distributed adversarial daemon from O(n^2) and O(delta m) to O(m) where n is the number of processes, m is thenumber of edges, and delta is the maximum degree in the graph

Investigations on push-relabel based algorithms for the maximum transversal problem

We investigate the push-relabel algorithm for solving the problem of finding a maximum cardinality matching in a bipartite graph in the context of the maximum transversal problem. We describe in detail an optimized yet easy-to-implement version of the algorithm and fine-tune its parameters. We also introduce new performance-enhancing techniques. On a wide range of real-world instances, we compare the push-relabel algorithm with state-of-the-art augmenting path-based algorithms and the recently proposed pseudoflow approach. We conclude that a carefully tuned push-relabel algorithm is competitive with all known augmenting path-based algorithms, and superior to the pseudoflow-based ones.Nous étudions le problème de couplage maximum dans des graphes bipartis. Nous décrivons en détail une version optimisée de l'algorithme en ajustant ses paramètres. L'algorithme est facile à mettre en œuvre. Nous introduisons également de nouvelles techniques pour améliorer la performance de l'algorithme. Sur un large éventail de cas du monde réel, nous comparons l'algorithme Push-Relabel avec des algorithmes basés sur les concepts de chemins augmentants et de pseudoflot récemment proposés. Nous concluons qu'un algorithme de type Push-Relabel soigneusement réglé est en concurrence avec tous les algorithmes connus de type chemins augmentants, et supérieur à ceux de type pseudoflot

A scalable parallel union-find algorithm for distributed memory computers

Abstract The Union-Find algorithm is used for maintaining a number of nonoverlapping sets from a finite universe of elements. The algorithm has applications in a number of areas including the computation of spanning trees and in image processing. Although the algorithm is inherently sequential there has been some previous efforts at constructing parallel implementations. These have mainly focused on shared memory computers. In this paper we present the first scalable parallel implementation of the Union-Find algorithm suitable for distributed memory computers. Our new parallel algorithm is based on an observation of how the Find part of the sequential algorithm can be executed more efficiently. We show the efficiency of our implementation through a series of tests to compute spanning forests of very large graphs