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Fast Algorithms for kShredders and kNode Connectivity Augmentation
, 1996
"... A kseparator (kshredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be knode connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counti ..."
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A kseparator (kshredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be knode connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of kseparators is #Pcomplete. However, we present an O(k )time (deterministic) algorithm for finding all the kshredders. This solves an open question: efficiently find a kseparator whose removal maximizes the number of connected 4, our running time is within a factor of k of the fastest algorithm known for testing knode connectivity. One application of shredders is in increasing the node connectivity from k to (k +1)by effi tly adding an (approximately) minimum number of new edges. Jord'an [JCT(B) 1995] gaveanO(n )time augmentation algorithm such that the number of new edges is within an additive term of (k 2) from a lower bound. We improve the running time to ), while achieving the same performance guarantee. For k 4, the running time compares favorably with the running time for testing knode connectivity.
NAEDGECONNECTIVITY AUGMENTATION PROBLEMS BY ADDING EDGES
, 2003
"... Abstract The network reliability in multiserver environment is measured by the connectivity between a vertex and a vertex subset (NAconnectivity). The problem of augmenting a graph by adding the smallest number of new edges to meet NAedge(vertex)connectivity requirement is an important optimizat ..."
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Abstract The network reliability in multiserver environment is measured by the connectivity between a vertex and a vertex subset (NAconnectivity). The problem of augmenting a graph by adding the smallest number of new edges to meet NAedge(vertex)connectivity requirement is an important optimization problem that contributes to the network design problem to increase the reliability of a current network by adding the smallest number of links. This problem is a generalization of the wellknown connectivity augmentation problems. In this paper, we focus on the NAedgeconnectivity augmentation problem. First, we prove the NPcompleteness of the problem which determines whether we can augment a graph to a 1NAedgeconnected graph by adding a given number or less new edges. Next, we prove that the problem of augmenting a 1NAedgeconnected graph or a 0NAedgeconnected graph to be 2NAedgeconnected graph by adding the smallest number of edges can be solved in polynomial time.
Connectivity Algorithms
"... This chapter presents an exposition of the advancement of the connectivity algorithms over the years. Most of these algorithms work by making a number of calls to a max‐flow subroutine. As these calls determine the bulk of the computation, attempts have been made to make the number of such calls as ..."
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This chapter presents an exposition of the advancement of the connectivity algorithms over the years. Most of these algorithms work by making a number of calls to a max‐flow subroutine. As these calls determine the bulk of the computation, attempts have been made to make the number of such calls as small as possible. 1.