In the Maximum Cut with Limited Unbalance problem, we want to partition the vertices of a weighted graph into two sets of sizes differing at most by a given threshold B, so that the sum of the weights of the crossing ...
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In the Maximum Cut with Limited Unbalance problem, we want to partition the vertices of a weighted graph into two sets of sizes differing at most by a given threshold B, so that the sum of the weights of the crossing edges is maximum. This problem has been introduced in (Galbiati and Maffioli, Theor Comput Sci 385 (2007), 78-87) where polynomial time randomized approximation algorithms are proposed and their performance guarantees are analyzed in the case of non-negative integer weights. In this article, we present extensive computational experience with these algorithms on a large number of different graphs. We then extend the analysis of these algorithms to integer weights not restricted in sign, and continue the computational testing. It turns out that the approximation ratios obtained are always substantially better than those guaranteed by the theoretical analysis. (C) 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 55(3), 247-255 2010
In the Maximum Cut with Limited Unbalance problem, we want to partition the vertices of a weighted graph into two sets of sizes differing at most by a given threshold B, so that the sum of the weights of the crossing ...
详细信息
In the Maximum Cut with Limited Unbalance problem, we want to partition the vertices of a weighted graph into two sets of sizes differing at most by a given threshold B, so that the sum of the weights of the crossing edges is maximum. This problem has been introduced in (Galbiati and Maffioli, Theor Comput Sci 385 (2007), 78-87) where polynomial time randomized approximation algorithms are proposed and their performance guarantees are analyzed in the case of non-negative integer weights. In this article, we present extensive computational experience with these algorithms on a large number of different graphs. We then extend the analysis of these algorithms to integer weights not restricted in sign, and continue the computational testing. It turns out that the approximation ratios obtained are always substantially better than those guaranteed by the theoretical analysis. (C) 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 55(3), 247-255 2010
We consider the problem of partitioning the vertices of a weighted graph into two sets of sizes that differ at most by a given threshold B, so as to maximize the weight of the crossing edges. For B equal to 0 this pro...
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We consider the problem of partitioning the vertices of a weighted graph into two sets of sizes that differ at most by a given threshold B, so as to maximize the weight of the crossing edges. For B equal to 0 this problem is known as Max Bisection, whereas for B equal to the number n of nodes it is the maximum cut problem. We present polynomial time randomized approximation algorithms with non trivial performance guarantees for its solution. The approximation results are obtained by extending the methodology used by Y. Ye for Max Bisection and by combining this technique with another one that uses the algorithm of Goemans and Williamson for the maximum cut problem. When B is equal to zero the approximation ratio achieved coincides with the one obtained by Y. Ye;otherwise it is always above this value and tends to the value obtained by Goemans and Williamson as B approaches the number n of nodes. (c) 2007 Elsevier B.V. All rights reserved.
The main focus of this paper is the analysis of a simple randomized scheme for constructing low-weight k-connected spanning subgraphs. In this paper, we focus on the metric graph. We use the term metric graph for a co...
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The main focus of this paper is the analysis of a simple randomized scheme for constructing low-weight k-connected spanning subgraphs. In this paper, we focus on the metric graph. We use the term metric graph for a complete graph with metric weights. We first show that our scheme gives a simple approximationalgorithm to construct a minimum-weight k-connected spanning subgraph in a metric graph, an NP-hard problem. We show that our algorithm gives an approximation ratio of O(k log n) for a metric graph, O(k) for a random graph with nodes uniformly randomly distributed in [0, 1](2) and O(log n/k) for a complete graph k with random edge weights U(0, 1). We show that our scheme is optimal with respect to the amount of "local information" needed to compute any connected spanning subgraph. We then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate k-connected spanning subgraph (for any k >= 1) in a point-to-point distributed model, where the processors form a complete network. Our algorithm takes O(log n/k) time and an expected number of O(nk log n/k) messages. Our result in conjunction with a result of Korach et al. [E. Korach, S. Moran, S. Zaks, The optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors, SIAM Journal on Computing 16 (2) (1987) 231-236] implies that the expected message complexity of our algorithm is significantly better than the best distributed algorithm that finds an optimal k-connected subgraph. We also show that for geometric instances, our randomized scheme constructs low-degree k-connected spanning subgraphs which have O(k log n) maximum degree, with high probability. (c) 2007 Elsevier B.V. All rights reserved.
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