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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.
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 Israel ..."
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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
Distributed Fractional Packing and Maximum Weighted bMatching via TailRecursive Duality
"... Abstract. We present efficient distributed δapproximation algorithms for fractional packing and maximum weighted bmatching in hypergraphs, where δ is the maximum number of packing constraints in which a variable appears (for maximum weighted bmatching δ is the maximum edge degree — for graphs δ = ..."
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Cited by 5 (2 self)
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Abstract. We present efficient distributed δapproximation algorithms for fractional packing and maximum weighted bmatching in hypergraphs, where δ is the maximum number of packing constraints in which a variable appears (for maximum weighted bmatching δ is the maximum edge degree — for graphs δ = 2). (a) For δ = 2 the algorithm runs in O(log m) rounds in expectation and with high probability. (b) For general δ, the algorithm runs in O(log 2 m) rounds in expectation and with high probability. 1 Background and results Given a weight vector w ∈ IR m +, a coefficient matrix A ∈ IR n×m and a vector b ∈ IR n +, the fractional packing problem is to compute a vector x ∈ IR m + to maximize ∑m j=1 wjxj and at the same time meet all the constraints ∑m j=1 Aijxj ≤ bi (∀i = 1... n). We use δ to denote the maximum number of packing constraints in which a variable appears, that is, δ = maxj {i  Aij ̸ = 0}. In the centralized setting, fractional packing
Distributed Algorithms for Covering, Packing and Maximum Weighted Matching
"... This paper gives polylogarithmicround, distributed δapproximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodularcost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mix ..."
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Cited by 4 (1 self)
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This paper gives polylogarithmicround, distributed δapproximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodularcost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives polylogarithmicround, distributed δapproximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted cMatching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms.
Bipartite graph matching computation on GPU
 in Proc. Intl. Conference Energy Minimization Methods in Computer Vision and Pattern Recognition
"... Abstract. The Bipartite Graph Matching Problem is a well studied topic in Graph Theory. Such matching relates pairs of nodes from two distinct sets by selecting a subset of the graph edges connecting them. Each edge selected has no common node as its end points to any other edge within the subset. W ..."
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Abstract. The Bipartite Graph Matching Problem is a well studied topic in Graph Theory. Such matching relates pairs of nodes from two distinct sets by selecting a subset of the graph edges connecting them. Each edge selected has no common node as its end points to any other edge within the subset. When the considered graph has huge sets of nodes and edges the sequential approaches are impractical, specially for applications demanding fast results. In this paper we investigate how to compute such matching on Graphics Processing Units (GPUs) motivated by its increasing processing power made available with decreasing costs. We present a new dataparallel approach for computing bipartite graph matching that is efficiently computed on today’s graphics hardware and apply it to solve the correspondence between 3D samples taken over a time interval. 1
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.