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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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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.
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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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
A Simpler Linear Time 2/3  ε Approximation for Maximum Weight Matching
 INF.PROCESS.LETT
"... We present two 2/3  ε approximation algorithms for the maximum weight matching problem that run in time O m log . We give a simple and practical randomized algorithm and a somewhat more complicated deterministic algorithm. Both algorithms are exponentially faster in terms of ..."
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We present two 2/3  &epsilon; approximation algorithms for the maximum weight matching problem that run in time O m log . We give a simple and practical randomized algorithm and a somewhat more complicated deterministic algorithm. Both algorithms are exponentially faster in terms of than a recent algorithm by Drake and Hougardy. We also show that our algorithms can be generalized to find a 1 approximatation to the maximum weight matching, for any > 0.
9 Linear Time Approximation Algorithms for Degree Constrained Subgraph Problems
"... Summary. Many realworld problems require graphs of such large size that polynomial time algorithms are too costly as soon as their runtime is superlinear. Examples include problems in VLSIdesign or problems in bioinformatics. For such problems the question arises: What is the best solution that ca ..."
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Summary. Many realworld problems require graphs of such large size that polynomial time algorithms are too costly as soon as their runtime is superlinear. Examples include problems in VLSIdesign or problems in bioinformatics. For such problems the question arises: What is the best solution that can be obtained in linear time? We survey linear time approximation algorithms for some classical problems from combinatorial optimization, e.g. matchings and branchings. For many combinatorial optimization problems arising from realworld applications, efficient, i.e., polynomial time algorithms are known for computing an optimum solution. However, there exist several applications for which the input size can easily exceed 10 9. In such cases polynomial time algorithms with a runtime that is quadratic
Maximum Weighted Matching Using the Partitioned Global Address Space Model
"... Efficient parallel algorithms for problems such as maximum weighted matching are central to many areas of combinatorial scientific computing. Manne and Bisseling [13] presented a parallel approximation algorithm which is well suited to distributed memory computers. This algorithm is based on a distr ..."
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Efficient parallel algorithms for problems such as maximum weighted matching are central to many areas of combinatorial scientific computing. Manne and Bisseling [13] presented a parallel approximation algorithm which is well suited to distributed memory computers. This algorithm is based on a distributed protocol due to Hoepman [9]. In the current paper, a partitioned global address space (PGAS) implementation is presented. PGAS programmers have the conveniences of using a shared memory model, which provides implicit communication between processes using normal loads and stores. Since the shared memory is partitioned according to the affinity of a process, one is also able to exploit data locality. This paper addresses the main differences between the PGAS and MPI implementations of the ManneBisseling algorithm. It highlights some advantages of using the PGAS model such as shorter, simpler code, similarity to the sequential algorithm, and options for finegrained and coarsegrained communication. 1.
Center Based Clustering: A Foundational Perspective
, 2013
"... In the first part of this chapter we present existing work in center based clustering methods. In particular, we focus on the kmeans and the kmedian clustering which are two of the most widely used clustering objectives. We describe popular heuristics for these methods and theoretical guarantees a ..."
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In the first part of this chapter we present existing work in center based clustering methods. In particular, we focus on the kmeans and the kmedian clustering which are two of the most widely used clustering objectives. We describe popular heuristics for these methods and theoretical guarantees associated with them. We also describe how to design worst case approximately optimal algorithms for these problems. In the second part of the chapter we describe recent work on how to improve on these worst case algorithms even further by using insights from the nature of real world clustering problems and data sets. Finally, we also summarize theoretical work on clustering data generated from mixture models such as a mixture of Gaussians. 1 Approximation algorithms for kmeans and kmedian One of the most popular approaches to clustering is to define an objective function over the data points and find a partitioning which achieves the optimal solution, or an approximately optimal solution to the given objective function. Common objective functions include center based objective functions such as kmedian and kmeans where one selects k center points and the clustering is obtained by assigning each data point to its closest center point. In
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, 2004
"... A simpler linear time 2/3 − ε approximation for maximum weight matching ..."
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Low Complexity, Stable Scheduling Algorithms for Networks of Input Queued Switches with No or Very Low Speedup
"... The delay and throughput characteristics of a packet switch depend mainly on the queueing scheme and the scheduling algorithm deployed at the switch. Early research on scheduling algorithms has mainly focused on maximum weight matching scheduling algorithms. It is known that maximum weight matchin ..."
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The delay and throughput characteristics of a packet switch depend mainly on the queueing scheme and the scheduling algorithm deployed at the switch. Early research on scheduling algorithms has mainly focused on maximum weight matching scheduling algorithms. It is known that maximum weight matching algorithms guarantee the stability of inputqueued switches, but are impractical due to their high computational complexity. Later research showed that the less complex maximal matching algorithms can stabilize inputqueued switches when they are deployed with a speedup of two. For practical purposes, neither a high computational complexity nor a speedup of two is desirable. In this paper, we investigate the application of matching algorithms that approximate maximum weight matching algorithms to scheduling problems. We show that while having a low computational complexity, they guarantee the stability of input queued switches when they are deployed with a moderate speedup. In particular, we show that the improve matching algorithm stabilizes inputqueued switches when it is deployed with a speedup of 3 2 + ǫ. In a second step, we further improve on these results by proposing a class of maximal weight matching algorithms that stabilize an inputqueued switch without any speedup. Whereas initial research has only focused on scheduling algorithms that guarantee the stability of a single switch, recent work has shown how scheduling algorithms for single switches can be modified in order to design distributed scheduling algorithms that stabilize networks of inputqueued switches. Using those results, we show that the switching algorithms proposed in this paper do not only stabilize a single switch, but also networks of inputqueued switches.