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Approximation Algorithms for Directed Steiner Problems
 Journal of Algorithms
, 1998
"... We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work we ..."
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Cited by 177 (8 self)
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We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i \Gamma 1)k 1=i in time O(n i k 2i ) for any fixed i ? 1, where k is the number of terminals. Thus, an O(k ffl ) approximation ratio can be achieved in polynomial time for any fixed ffl ? 0. Setting i = log k, we obtain an O(log 2 k) approximation ratio in quasipolynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k 2=3 log 1=3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner...
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
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Hardness of buyatbulk network design
 In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
, 2004
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On the RedBlue Set Cover Problem
 In Proceedings of the 11th Annual ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B , the redblue set cover problem is to find a subfamily C ` S which covers all blue elements, but which covers the minimum possible number of red elements. We note that RedBlue Se ..."
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Cited by 48 (0 self)
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Given a finite set of "red" elements R, a finite set of "blue" elements B and a family S ` 2 R[B , the redblue set cover problem is to find a subfamily C ` S which covers all blue elements, but which covers the minimum possible number of red elements. We note that RedBlue Set Cover is closely related to several combinatorial optimization problems studied earlier. These include the group Steiner problem, directed Steiner problem, minimum label path, minimum monotone satisfying assignment and symmetric label cover. From the equivalence of RedBlue Set Cover and MMSA3 it follows that, unless P=NP, even the restriction of RedBlue Set Cover where every set contains only one blue and two red elements cannot be approximated to within O(2 log 1\Gammaffi n ) , where ffi = 1= log log c n, for any constant c ! 1=2 (where n = S). We give integer programming formulations of the problem and use them to obtain a 2 p n approximation algorithm for the restricted case of RedBlue Set Cove...
Transitiveclosure spanners
, 2008
"... We define the notion of a transitiveclosure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. These spanner ..."
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Cited by 38 (11 self)
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We define the notion of a transitiveclosure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a ktransitiveclosurespanner (kTCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitiveclosure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property testing, and data structures, and properties of these spanners have been rediscovered over the span of 20 years. We bring these areas under the unifying framework of TCspanners. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TCspanners. We study the approximability of the size of the sparsest kTCspanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs,
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 25 (13 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
Treedecompositions with bags of small diameter
, 2007
"... This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right ..."
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Cited by 19 (1 self)
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This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded treelength graphs includes (but is not reduced to) bounded chordality graphs (like interval graphs, permutation graphs, ATfree graphs, etc.). For instance, we show that the treelength of any outerplanar graph is ⌈k/3⌉, where k is the chordality of the graph, and we compute the treelength of meshes. More fundamentally we show that any algorithm computing a treedecomposition approximating the treewidth (or the treelength) of an nvertex graph by a factor α or less does not give an αapproximation of the treelength (resp. the treewidth) unless if α = Ω(n 1/5). We complete these results presenting several polynomial time constant approximate algorithms for the treelength. The introduction of this parameter is motivated by the design of compact distance labeling, compact routing tables with nearoptimal route length, and by the construction of sparse additive spanners.
Set Connectivity Problems in Undirected Graphs and the Directed Steiner Network Problem
"... In the generalized connectivity problem, we are given an edgeweighted graph G = (V, E) and a collection D = {(S1, T1),..., (Sk, Tk)} of distinct demands; each demand (Si, Ti) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand (Si, Ti) when it contains a path with ..."
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Cited by 19 (3 self)
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In the generalized connectivity problem, we are given an edgeweighted graph G = (V, E) and a collection D = {(S1, T1),..., (Sk, Tk)} of distinct demands; each demand (Si, Ti) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand (Si, Ti) when it contains a path with one endpoint in Si and the other in Ti. The goal is to identify a minimum weight subgraph that connects all demands in D. Alon et al. (SODA ’04) introduced this problem to study online network formation settings and showed that it captures some wellstudied problems such as Steiner forest, facility location with nonmetric costs, tree multicast, and group Steiner tree. Finding a nontrivial approximation ratio for generalized connectivity was left as an open problem. We describe the first polylogarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O(log 2 n log 2 k). Here, n is the number of vertices in G and k is the number of demands. We also prove that the cutcovering relaxation of this problem has an O(log 3 n log 2 k) integrality gap. Building upon the results for generalized connectivity, we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the
The Hardness of Approximating Spanner Problems
"... This paper examines a number of variants of the sparse kspanner problem, and presents hardness results concerning their approximability. Previously, it was known that most kspanner problems are weakly inapproximable, namely, are NPhard to approximate with ratio O(log n), for every k * 2, and that ..."
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Cited by 17 (2 self)
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This paper examines a number of variants of the sparse kspanner problem, and presents hardness results concerning their approximability. Previously, it was known that most kspanner problems are weakly inapproximable, namely, are NPhard to approximate with ratio O(log n), for every k * 2, and that the unitlength kspanner problem for constant stretch requirement k * 5 is strongly inapproximable, namely, is NPhard to approximate with ratio O(2log ffln) [19].