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Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
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FLSS: A FaultTolerant Topology Control Algorithm for Wireless Networks
, 2004
"... Topology control algorithms usually reduce the number of links in a wireless network, which in turn decreases the degree of connectivity. The resulting network topology is more susceptible to system faults such as node failures and departures. In this paper, we consider kvertex connectivity of a wi ..."
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Cited by 71 (4 self)
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Topology control algorithms usually reduce the number of links in a wireless network, which in turn decreases the degree of connectivity. The resulting network topology is more susceptible to system faults such as node failures and departures. In this paper, we consider kvertex connectivity of a wireless network. We first present a centralized algorithm, Faulttolerant Global Spanning Subgraph (FGSSk), which preserves kvertex connectivity. FGSSk is minmax optimal, i.e., FGSSk minimizes the maximum transmission power used in the network, among all algorithms that preserve kvertex connectivity. Based on FGSSk, we propose a localized algorithm, Faulttolerant Local Spanning Subgraph (FLSSk). It is proved that FLSSk preserves kvertex connectivity while maintaining bidirectionality of the network, and FLSSk is minmax optimal among all strictly localized algorithms. We then relax several widely used assumptions for topology control to enhance the practicality of FGSSk and FLSSk. Simulation results show that FLSSk is more powerefficient than other existing distributed/localized topology control algorithms.
Hardness of Approximation for VertexConnectivity NetworkDesign Problems
, 2002
"... In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths con ..."
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Cited by 50 (4 self)
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In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths connecting them.
Approximating MinimumSize kConnected Spanning Subgraphs via Matching
 SIAM J. Comput
, 1998
"... Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spann ..."
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Cited by 43 (3 self)
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Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spanning subgraph of an undirected graph 1+[1=k], minimumsize knode connected spanning subgraph of a directed graph 1+[1=k], minimumsize kedge connected spanning subgraph of an undirected graph 1+[2=(k + 1)], and minimumsize kedge connected spanning subgraph of a directed graph 1+[4= p k].
A 2approximation algorithm for finding an optimum 3vertexconnected spanning subgraph
 Journal of Algorithms
, 1999
"... The problem of finding a minimum weight kvertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NPhard. Combining properties of inclusionminimal kvertex connected graphs and of koutconnected graphs (i.e., graphs which contain a vertex from ..."
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Cited by 35 (13 self)
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The problem of finding a minimum weight kvertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NPhard. Combining properties of inclusionminimal kvertex connected graphs and of koutconnected graphs (i.e., graphs which contain a vertex from which there exist k internally vertexdisjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( ⌈ k ⌉ + 1)connected subgraph 2 with a weight at most twice the optimum to the original problem. In particular, we obtain a 2approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(V  3 E) = O(V  5).
Approximating Steiner Networks with Node Weights
, 2007
"... The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uvedgeconnectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jai ..."
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Cited by 26 (12 self)
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The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uvedgeconnectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jain [19], and numerous papers preceding it, considered the EdgeWeighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum weight edgecovers of several types of set functions and families. However, for the NodeWeighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation, by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(ln U), where rmax = maxu,v∈U r(u, v). This generalizes the result of Klein and Ravi [21] for the case rmax = 1. We also give an O(ln U)approximation algorithm for the nodeconnectivity variant of NWSN (when the paths are required to be internallydisjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum nodeweighted edgecover of an uncrossable setfamily. We also give the first evidence that a polylogarithmic approximation ratio for NWSN might not exist even for U  = 2 and unit weights. 1 1
On Approximability of the MinimumCost kConnected Spanning Subgraph Problem
, 1999
"... We present the first truly polynomialtime approximation scheme (PTAS) for the minimumcost kvertex (or, k edge) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " how to construct in a polynomial time a graph o ..."
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Cited by 25 (7 self)
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We present the first truly polynomialtime approximation scheme (PTAS) for the minimumcost kvertex (or, k edge) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " how to construct in a polynomial time a graph on a superset of the input points which is kvertex connected with respect to the input points, and whose cost is within (1+ ") of the minimumcost of a kvertex connected graph spanning the input points. We subsume that result by showing for every positive constant " how to construct in a polynomialtime a kconnected subgraph spanning the input points without any Steiner points and having the cost within (1 + ") of the minimum. We also study hardness of approximations for the minimumcost kvertex and kedgeconnected spanning subgraph problems. The only inapproximability result known so far for the minimumcost kvertex and kedge connected spanning subgraph problems states that the k edge...
On 2Coverings and 2Packings of Laminar Families
, 1999
"... . Let H be a laminar family of subsets of a groundset V . A kcover of H is a multiset C of edges on V such that for every subset S in H, C has at least k edges that have exactly one end in S. A kpacking of H is a multiset P of edges on V such that for every subset S in H, P has at most k \Delt ..."
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Cited by 19 (2 self)
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. Let H be a laminar family of subsets of a groundset V . A kcover of H is a multiset C of edges on V such that for every subset S in H, C has at least k edges that have exactly one end in S. A kpacking of H is a multiset P of edges on V such that for every subset S in H, P has at most k \Delta u(S) edges that have exactly one end in S. Here, u assigns an integer capacity to each subset in H. Our main results are: (a) Given a kcover C of H, there is an efficient algorithm to find a 1cover contained in C of size kjCj=(2k \Gamma 1). For 2covers, the factor of 2=3 is best possible. (b) Given a 2packing P of H, there is an efficient algorithm to find a 1packing contained in P of size jP j=3. The factor of 1=3 for 2packings is best possible. These results are based on efficient algorithms for finding appropriate colorings of the edges in a kcover or a 2packing, respectively, and they extend to the case where the edges have nonnegative weights. Our results imply app...
Approximating connectivity augmentation problems
 In SODA. 176–185
, 2005
"... Let G = (V, E) be an undirected graph and let S ⊆ V. The Sconnectivity λS G (u, v) of a node pair (u, v) in G is the maximum number of uvpaths that no two of them have an edge or a node in S − {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = (V, E), ..."
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Cited by 17 (11 self)
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Let G = (V, E) be an undirected graph and let S ⊆ V. The Sconnectivity λS G (u, v) of a node pair (u, v) in G is the maximum number of uvpaths that no two of them have an edge or a node in S − {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = (V, E), a node subset S ⊆ V, and a nonnegative integer requirement function r(u, v) on V ×V, add a minimum size set F of new edges to G so that λS G+F (u, v) ≥ r(u, v) for all (u, v) ∈ V ×V. Three extensively studied particular cases are: the EdgeCA (S = ∅), the NodeCA (S = V), and the ElementCA (r(u, v) = 0 whenever u ∈ S or v ∈ S). A polynomial algorithm for EdgeCA was developed by Frank. In this paper we consider the ElementCA and the NodeCA, that are NPhard even for r(u, v) ∈ {0, 2}. The best known ratios for these problems were: 2 for ElementCA and O(rmax · ln n) for NodeCA, where rmax = maxu,v∈V r(u, v) and n = V . Our main result is a 7/4approximation algorithm for the ElementCA, improving the previously best known 2approximation. For ElementCA with r(u, v) ∈ {0, 1, 2} we give a 3/2approximation algorithm. These approximation ratios are based on a new splittingoff theorem, which implies an improved lower bound on the number of edges needed to cover a skewsupermodular set function. For NodeCA we establish the following approximation threshold: NodeCA with r(u, v) ∈ {0, k} cannot be approximated within O(2log1−ε n) for any fixed ε> 0, unless NP ⊆ DTIME(npolylog(n)).