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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 th ..."
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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 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.
Algebraic Distance on Graphs
"... Measuring the connection strength between a pair of vertices in a graph is one of the most vital concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the local connectivity. In this paper, we consider a neighborhood of each graph vertex and pr ..."
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Measuring the connection strength between a pair of vertices in a graph is one of the most vital concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the local connectivity. In this paper, we consider a neighborhood of each graph vertex and propagate a certain property value through direct neighbors. We present a measure of the connection strength (called the algebraic distance, see [21]) defined from an iterative process based on this consideration. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. A rigorous analysis of the convergence property of the process confirms the underlying intuition that vertices are mutually reinforced and that the local neighborhoods play an important role in influencing the vertex connectivity. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs. 1
Scaling algorithms for approximate and exact maximum weight matching
, 2011
"... The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the ..."
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The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the following sense. For any ɛ> 0, we give an algorithm that computes a (1 − ɛ)approximate maximum weight matching in O(mɛ −1 log ɛ −1) time, that is, optimal linear time for any fixed ɛ. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error. Our second contribution is a new exact maximum weight matching algorithm for integerweighted bipartite graphs that runs in time O(m √ n log N). This improves on the O(Nm √ n)time and O(m √ n log(nN))time algorithms known since the mid 1980s, for 1 ≪ log N ≪ log n. Here N is the maximum integer edge weight. 1
Computing steiner minimum trees in Hamming metric
 In Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm
, 2006
"... Computing Steiner minimum trees in Hamming metric is a well studied problem that has applications in several fields of science such as computational linguistics and computational biology. Among all methods for finding such trees, algorithms using variations of a branch and bound method developed by ..."
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Computing Steiner minimum trees in Hamming metric is a well studied problem that has applications in several fields of science such as computational linguistics and computational biology. Among all methods for finding such trees, algorithms using variations of a branch and bound method developed by Penny and Hendy have been the fastest for more than 20 years. In this paper we describe a new pruning approach that is superior to previous methods and its implementation. 1
Maximizing Cooperative Diversity Energy Gain for Wireless Networks
, 2007
"... We are concerned with optimally grouping active mobile users in a twouserbased cooperative diversity system to maximize the cooperative diversity energy gain in a radio cell. The optimization problem is formulated as a nonbipartite weightedmatching problem in a static network setting. The weighte ..."
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We are concerned with optimally grouping active mobile users in a twouserbased cooperative diversity system to maximize the cooperative diversity energy gain in a radio cell. The optimization problem is formulated as a nonbipartite weightedmatching problem in a static network setting. The weightedmatching problem can be solved using maximum weighted (MW) matching algorithm in polynomial time O(n 3). To reduce the implementation and computational complexity, we develop a WorstLinkFirst (WLF) matching algorithm, which gives the user with the worse channel condition and the higher energy consumption rate a higher priority to choose its partner. The computational complexity of the proposed WLF algorithm is O(n 2) while the achieved average energy gain is only slightly lower than that of the optimal maximum weightedmatching algorithm and similar to that of the 1/2approximation Greedy matching algorithm (with computational complexity of O(n 2 log n)) for a staticuser network. We further investigate the optimal matching problem in mobile networks. By intelligently applying user mobility information in the matching algorithm, high cooperative diversity energy gain with moderate overhead is possible. In mobile networks, the proposed WLF matching algorithm, being less complex than the MW and the Greedy matching algorithms, yields performance characteristics close to those of the MW matching algorithm and better than the Greedy matching algorithm.
A Simple Parallel Approximation Algorithm for the Weighted Matching Problem
, 2007
"... Given a weighted graph, the weighted matching problem is to find a matching with maximum weight. The fastest known exact algorithm runs in O(nm + n2 log n) however for many real world applications this is too costly, and an approximate matching is sufficient. A capproximation algorithm is one which ..."
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Given a weighted graph, the weighted matching problem is to find a matching with maximum weight. The fastest known exact algorithm runs in O(nm + n2 log n) however for many real world applications this is too costly, and an approximate matching is sufficient. A capproximation algorithm is one which always finds a weight of at least c times the optimal weight. Drake and Hougardy developed a linear time 2/3  epsilon approximation algorithm which is the best known serial algorithm. They also developed a parallel 1 epsilon approximation algorithm for the PRAM model, however it requires a large number of processors which is not as useful in practice. Hoepman developed a distributed 1/2 approximation algorithm which is the best known distributed algorithm. We present a shared memory parallel version of the best 2/3  epsilon algorithm, which is simple to understand and easy to implement.
Monograph
"... The performance of wireless networks depends fundamentally on the characteristics of the radio resource. In this thesis we study methods that can be used to improve performance of wireless networks. We also study methods that can be used to analyze the performance of such networks. In the first part ..."
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The performance of wireless networks depends fundamentally on the characteristics of the radio resource. In this thesis we study methods that can be used to improve performance of wireless networks. We also study methods that can be used to analyze the performance of such networks. In the first part of the thesis, we propose algorithms for multicast routing and maxmin fair link scheduling in wireless multihop networks. The multicast routing problem is to find a minimumcost sequence of transmissions which delivers a message from a given source node to a set of destination nodes. We propose three efficient multicast routing algorithms for certain common instances of the problem. The first algorithm assumes fixed tranmission costs and constructs an efficient multicast tree in a centralized fashion.
ParComb  Parallel Algorithms for Combinatorial Scientific Computing
"... Although scientific computing is traditionally viewed as the province of continuous mathematics, of differential equations and linear algebra, there are many combinatorial subproblems that arise in the solution of scientific computing problems. Research on these types of problems has been carried ou ..."
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Although scientific computing is traditionally viewed as the province of continuous mathematics, of differential equations and linear algebra, there are many combinatorial subproblems that arise in the solution of scientific computing problems. Research on these types of problems has been carried out for several years but it is only recently that it has been recognized as a field of its own, now known as “Combinatorial scientific computing” (CSC). This is an interdisciplinary research area involving discrete mathematics in scientific computing and refers to the development, analysis, and application of combinatorial algorithms to solve problems in computational science and engineering. The importance and interest in the field is mirrored by the number of conferences and workshops dedicated to CSC that has been arranged over the last years. As an example of the role that CSC plays in scientific computing consider the problem of computing a matrixvector product y = Ax where A is large and sparse. This is the core operation of almost every iterative equation solver. When this operation is carried out on a parallel computer, as is often the case for large applications, it has long been recognized that one should partition A so that each processor gets an almost equal share of the data elements while at the same time minimizing the amount of required communication. Achieving such a partitioning is a pure combinatorial problem. By viewing the matrix A as either a graph or a hypergraph researchers have developed a number of
A Measure of the Connection Strengths between Graph Vertices with Applications
, 909
"... We present a simple iterative strategy for measuring the connection strength between a pair of vertices in a graph. The method is attractive in that it has a linear complexity and can be easily parallelized. Based on an analysis of the convergence property, we propose a mutually reinforcing model to ..."
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We present a simple iterative strategy for measuring the connection strength between a pair of vertices in a graph. The method is attractive in that it has a linear complexity and can be easily parallelized. Based on an analysis of the convergence property, we propose a mutually reinforcing model to explain the intuition behind the strategy. The practical effectiveness of this measure is demonstrated through several combinatorial optimization problems on graphs and hypergraphs. 1
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 Fast Combinatorial Vector Field Topology
"... Abstract—This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman’s work on a discrete Morse theory for dynamical systems. A computational f ..."
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Abstract—This paper introduces a novel approximation algorithm for the fundamental graph problem of combinatorial vector field topology (CVT). CVT is a combinatorial approach based on a sound theoretical basis given by Forman’s work on a discrete Morse theory for dynamical systems. A computational framework for this mathematical model of vector field topology has been developed recently. The applicability of this framework is however severely limited by the quadratic complexity of its main computational kernel. In this work we present an approximation algorithm for CVT with a significantly lower complexity. This new algorithm reduces the runtime by several orders of magnitude, and maintains the main advantages of CVT over the continuous approach. Due to the simplicity of our algorithm it can be easily parallelized to improve the runtime further.