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52
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
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Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems
"... A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞ ..."
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Cited by 61 (1 self)
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A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of nonzero coefficients in a row of U, and provide a polynomialtime approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first nontrivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomialtime approximation algorithms to match this bound. This improves on the best previous mapproximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentiallymany) inequalities added. Our contribution here is to describe an appropriate
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 44 (5 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 35 (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 33 (15 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).
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 on a s ..."
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Cited by 20 (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...
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 20 (15 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 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 16 (3 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 15 (12 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)).
A Uniform Framework for Approximating Weighted Connectivity Problems
 In Proc. 10th ACMSIAM Annual Symposium on Discrete Algorithms
, 1999
"... We introduce a new algorithmic technique that applies to several graph connectivity problems. Its power is demonstrated by experimental studies of the minimumweight stronglyconnected spanning subgraph problem and the minimumweight augmentation problem. Even though we are unable to improve the app ..."
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Cited by 13 (1 self)
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We introduce a new algorithmic technique that applies to several graph connectivity problems. Its power is demonstrated by experimental studies of the minimumweight stronglyconnected spanning subgraph problem and the minimumweight augmentation problem. Even though we are unable to improve the approximation ratios for these problems, our studies indicate that the new method generates significantly better solutions than the current known approximation algorithms, and yields solutions very close to optimal. We believe that our technique will eventually lead to algorithms that improve the performance ratios as well. 1 Introduction Let a weighted graph G = (V; E) represent all the feasible links of a potential communications network. A minimum spanning tree in G is the cheapest connected subgraph, i.e., the cheapest network that will allow the sites to communicate. Such a network is highly susceptible to failures, since it cannot even survive a single link or site failure. For more rel...