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49
Engineering a Scalable High Quality Graph Partitioner
 24th IEEE International Parallal and Distributed Processing Symposium (IPDPS
, 2010
"... We describe an approach to parallel graph partitioning that scales to hundreds of processors and produces a high solution quality. For example, for many instances from Walshaw’s benchmark collection we improve the best known partitioning. We use the well known framework of multilevel graph partiti ..."
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Cited by 31 (17 self)
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We describe an approach to parallel graph partitioning that scales to hundreds of processors and produces a high solution quality. For example, for many instances from Walshaw’s benchmark collection we improve the best known partitioning. We use the well known framework of multilevel graph partitioning. All components are implemented by scalable parallel algorithms. Quality improvements compared to previous systems are due to better prioritization of edges to be contracted, better approximation algorithms for identifying matchings, better local search heuristics, and perhaps most notably, a parallelization of the FM local search algorithm that works more locally than previous approaches. 1
A Linear Time Approximation Algorithm for Weighted Matchings in Graphs
, 2003
"... 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 p ..."
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Cited by 25 (3 self)
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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
Greedy in Approximation Algorithms
 PROC. OF ESA
, 2006
"... The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of kextendible systems, a natural generalization of matroids, and show that a greedy algorithm is a 1factor approximatio ..."
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Cited by 23 (1 self)
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The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of kextendible systems, a natural generalization of matroids, and show that a greedy algorithm is a 1factor approximation for these systems. Many seemly unrelated k problems fit in our framework, e.g.: bmatching, maximum profit scheduling and maximum asymmetric TSP. In the second half of the paper we focus on the maximum weight bmatching problem. The problem forms a 2extendible system, so greedy gives us a 1factor solution which runs in 2 O(m log n) time. We improve this by providing two linear time approximation algorithms for the problem: a 1 2factor algorithm that runs in O(bm) time, and a `2 3 − ǫ ´factor algorithm which runs in expected O ` bm log 1 ´ time.
A lineartime approximation algorithm for weighted matchings in graphs
 ACM TRANSACTIONS ON ALGORITHMS
, 2005
"... 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 p ..."
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Cited by 22 (0 self)
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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 polynomialtime 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 + n² 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 lineartime approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/1. This improves the previously best performance ratio of 3/2. Our algorithm is not only of theoretical interest, but because it is easy to implement and the constants involved are quite small it is also useful in practice.
Simple distributed weighted matchings
 In eprint cs.DC/0410047
, 2004
"... Wattenhofer et al. [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 comput ..."
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Cited by 21 (1 self)
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Wattenhofer et al. [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. 1
Transport in dynamical astronomy and multibody problems
 INT. J. OF BIFURCATION AND CHAOS
, 2005
"... We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resona ..."
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Cited by 21 (8 self)
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We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.
Improved Distributed Approximate Matching
"... We present improved algorithms for finding approximately optimal matchings in both weighted and unweighted graphs. For unweighted graphs, we give an algorithm providing (1 − ɛ)approximation in O(log n) time for any constant ɛ> 0. This result improves on the classical 1approximation due 2 to Isr ..."
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Cited by 20 (3 self)
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We present improved algorithms for finding approximately optimal matchings in both weighted and unweighted graphs. For unweighted graphs, we give an algorithm providing (1 − ɛ)approximation in O(log n) time for any constant ɛ> 0. This result improves on the classical 1approximation due 2 to Israeli and Itai. As a byproduct, we also provide an improved algorithm for unweighted matchings in bipartite graphs. In the context of weighted graphs, we give another algorithm which provides ( 1 − ɛ) approximation in general 2 graphs in O(log n) time. The latter result improves on the − ɛ)approximation in O(log n) time. known ( 1 4
Approximating Maximum Weight Matching in Nearlinear Time
"... Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertexdisjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m √ n) time, where m and n ..."
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Cited by 20 (3 self)
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Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertexdisjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m √ n) time, where m and n are the number of edges and vertices in the graph. Surprisingly, restricted versions of the problem, such as computing (1 − ɛ)approximate MWMs or finding maximum cardinality matchings, are not known to be much easier (on sparse graphs). The best algorithms for these problems also run in Õ(m √ n) time. In this paper we present the first nearlinear time algorithm for computing (1 − ɛ)approximate MWMs. Specifically, given an arbitrary realweighted graph and ɛ> 0, our algorithm computes such a matching in O(mɛ −2 log 3 n) time. The previous best approximate MWM algorithm with comparable running time could only guarantee a (2/3 − ɛ)approximate solution. In addition, we present a faster algorithm, running in O(m log n log ɛ −1) time, that computes a (3/4−ɛ)approximate MWM.
A parallel approximation algorithm for the weighted maximum matching problem
 In Proc. Seventh Int. Conf. on Parallel Processing and Applied Mathematics (PPAM
, 2007
"... Abstract. We consider the problem of computing a weighted edge matching in a large graph using a parallel algorithm. This problem has application in several areas of combinatorial scientific computing. Since an exact algorithm for the weighted matching problem is both fairly expensive to compute and ..."
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Abstract. We consider the problem of computing a weighted edge matching in a large graph using a parallel algorithm. This problem has application in several areas of combinatorial scientific computing. Since an exact algorithm for the weighted matching problem is both fairly expensive to compute and hard to parallelise we instead consider fast approximation algorithms. We analyse a distributed algorithm due to Hoepman [8] and show how this can be turned into a parallel algorithm. Through experiments using both complete as well as sparse graphs we show that our new parallel algorithm scales well using up to 32 processors. 1
Engineering algorithms for approximate weighted matching
 In Proceedings of the 6th International Workshop on Experimental Algorithms
, 2007
"... Abstract. We present a systematic study of approximation algorithms for the maximum weight matching problem. This includes a new algorithm which provides the simple greedy method with a recent path heuristic. Surprisingly, this quite simple algorithm performs very well, both in terms of running time ..."
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Cited by 17 (1 self)
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Abstract. We present a systematic study of approximation algorithms for the maximum weight matching problem. This includes a new algorithm which provides the simple greedy method with a recent path heuristic. Surprisingly, this quite simple algorithm performs very well, both in terms of running time and solution quality, and, though some other methods have a better theoretical performance, it ranks among the best algorithms. 1