Engineering Fast Parallel Matching Algorithms

The computation of matchings has applications in the solving process of a large variety of problems, e.g. as part of graph partitioners. We present and analyze three sequential and two parallel approximation algorithms for the cardinality and weighted matching problem. One of the sequential algorithms is based on an algorithm by Karp and Sipser to compute maximal matchings [21]. Another one is based on the idea of locally heaviest edges by Preis [30]. The third sequential algorithm is a new algorithm based on the computation of maximum weighted matchings of trees spanning the input graph. We show for two of these algorithms that the runtime for slight variations of them is expected to be linear. However the experimental results suggest that this is also the case for the unmodified versions. The comparison with other approximate matching algorithms show that the computed matchings have a similar quality or even the same quality. On the other hand two of our the algorithms are much faster. For two of the sequential algorithms we show how to turn them into parallel matching algorithms. We show that for a simple non optimal partitioning of the input graphs speedups can be observed using up to 1024 processors. For certain kinds of input graphs we see a good scaling behaviour

Algoritmos para emparelhamentos em grafos bipartidos

Orientador : Claudio Leonardo LucchesiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Ciencia da ComputaçãoResumo: O problema de emparelhamentos em grafos consiste em determinar um conjunto M de arestas do grafo, onde as arestas são disjuntas nos vértices. Em particular, estamos interessados em determinar emparelhamentos máximos, ou seja, de cardinalidade máxima. Existem muitas variações em torno do tema, o grafo pode ser: bipartido ou não, ponderado ou não. Neste trabalho apresentamos as principais técnicas para se projetar os algoritmos mais eficientes que resolvem o problema de emparelhamentos máximos, ponderados ou não, em grafos bipartidos. Também descrevemos os principais algoritmos, seqüenciais e paralelos, que resolvem este problema. O Capítulo 2 apresenta os principais algoritmos para resolver o problema em grafos bipartidos não ponderados: o algoritmo de Hopcroft e Karp, o algoritmo paralelo de Kim e Chwa e o algoritmo paralelo de Goldberg, Plotkin e Vaidya. O Capítulo 3 apresenta os principais algoritmos para resolver o problema em grafos bipartidos ponderados: o algoritmo de Edmonds e Karp, o algoritmo com escalonamento de Gabow, o algoritmo com escalonamento e aproximação de Gabow e Tarjan, o algoritmo paralelo de Goldberg, Plotkin e Vaidya e o algoritmo paralelo de Gabow e Tarjan. O Apêndice A contém uma tabela dos principais algoritmos para resolver o problema no caso em que os grafos não são bipartidosAbstract: The matching problem in graphs consists in determining a vertex disjoint set M of edges of the graph. In particular, we are interested in finding maximum matchings, that is, matchings of maximum cardinality. There are many variations around this problem, the graph can be: bipartite or general, weighted or not. In this work we present the main techniques to design the most efficient algorithms that solve the problem of maximum matching, weighted or not, in bipartite graphs. We also describe the main algorithms, sequential and parallel, to solve this problem. Chapter 2 contains the most important algorithms to solve the problem for non weighted bipartite graphs, namely, the algorithm of Hopcroft and Karp, the parallel algorithm of Kim and Chwa, and the parallel algorithm of Goldberg, Plotkin and Vaidya. Chapter 3 contains the most important algorithms to solve the problem for weighted bipartite graphs, namely, the algorithm of Edmonds and Katp, the scaling algorithm of Gabow, the scaling and approximation algorithm of Gabow and Tarjan, the parallel algorithm of Goldberg, Plotkin and Vaidya and the parallel algorithm of Gabow and Tarjan. In Appendix A it is given a table which describes briefly the most important algorithms for solving the general problem, in which the graph is not bipartiteMestradoMestre em Ciência da Computaçã

Simple Distributed Weighted Matchings

Wattenhofer [WW04] derive a complicated distributed algorithm to compute a weighted matching of an arbitrary weighted graph, that is at most a factor 5 away from the maximum weighted matching of that graph. We show that a variant of the obvious sequential greedy algorithm [Pre99], that computes a weighted matching at most a factor 2 away from the maximum, is easily distributed. This yields the best known distributed approximation algorithm for this problem so far

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is $1/2$, which is achieved by any greedy algorithm. D\"urr et al. recently presented a $2$-pass algorithm called Category-Advice that achieves approximation ratio $3/5$. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the $k$-pass Category-Advice algorithm for all $k \ge 1$, and show that the approximation ratio converges to the inverse of the golden ratio $2/(1+\sqrt{5}) \approx 0.618$ as $k$ goes to infinity. The convergence is extremely fast --- the $5$-pass Category-Advice algorithm is already within $0.01\%$ of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than $1-1/e$, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model

Linear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem

In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph. In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniques to adapt and optimize linear programming based approaches in the semi-streaming model with an application to the maximum matching problem. As a consequence, we improve (almost) all previous results on this problem, and also prove new results on interesting variants

Best of Two Local Models: Local Centralized and Local Distributed Algorithms

We consider two models of computation: centralized local algorithms and local distributed algorithms. Algorithms in one model are adapted to the other model to obtain improved algorithms. Distributed vertex coloring is employed to design improved centralized local algorithms for: maximal independent set, maximal matching, and an approximation scheme for maximum (weighted) matching over bounded degree graphs. The improvement is threefold: the algorithms are deterministic, stateless, and the number of probes grows polynomially in $\log^* n$, where $n$ is the number of vertices of the input graph. The recursive centralized local improvement technique by Nguyen and Onak~\cite{onak2008} is employed to obtain an improved distributed approximation scheme for maximum (weighted) matching. The improvement is twofold: we reduce the number of rounds from $O(\log n)$ to $O(\log^*n)$ for a wide range of instances and, our algorithms are deterministic rather than randomized