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17
Approximation Algorithms for MinimumCost kVertex Connected Subgraphs
 In 34th Annual ACM Symposium on the Theory of Computing
, 2002
"... We present two new algorithms for the problem of nding a minimumcost kvertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works o ..."
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Cited by 72 (2 self)
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We present two new algorithms for the problem of nding a minimumcost kvertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works on any graph (directed or undirected) and gives an O( n=)approximation algorithm for any > 0 and k (1 )n. These algorithms improve on the previous best approximation factor (more than k=2). The latter algorithm also extends to other problems in network design with vertex connectivity requirements. Our main tools are setpair relaxations, a theorem of Mader's (in the undirected case) and iterative rounding (general case).
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 52 (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.
Iterative Rounding 2Approximation Algorithms for MinimumCost Vertex Connectivity Problems
 J. Comput. Syst. Sci
, 2002
"... The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j, find a minimumcost subgraph such that there are r ij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP) ..."
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Cited by 44 (0 self)
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The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j, find a minimumcost subgraph such that there are r ij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP) , these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. The element connectivity problem (ELCSNDP, or ELC) is a problem of intermediate difficulty.
An Iterative Rounding 2Approximation Algorithm for the Element Connectivity Problem
 In 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... In the edge connected version of this problem (ECSNDP), these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element conne ..."
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Cited by 26 (2 self)
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In the edge connected version of this problem (ECSNDP), these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELCSNDP, or ELC). In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values are only specified for pairs of terminals must be element disjoint. Thus if are still connected by a path in the network. These variants of SNDP are all known to be NPhard. The best known approximation algorithm for the ECSNDP has performance guarantee of 2 (due to Jain [11]), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primaldual  approximation algorithm, where (Jain et al. [12]). VCSNDP is not known to have a nontrivial approximation algorithm; however, recently Fleischer [7] has shown how to extend the technique of Jain [11] to give a 2approximation algorithm in the case that ! . She also shows that the same techniques will not work for VCSNDP for more general values of . In this paper we show that these techniques can be extended to a 2approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures.
A graph reduction step preserving elementconnectivity and applications
 IN INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2009
"... Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the elementconnectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of uv paths that are pairwise disjoint in both edges and nonterminals V \ T (the paths need not be disjoint in terminals). Elementconnectivity ..."
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Cited by 10 (2 self)
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Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the elementconnectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of uv paths that are pairwise disjoint in both edges and nonterminals V \ T (the paths need not be disjoint in terminals). Elementconnectivity is more general than edgeconnectivity and less general than vertexconnectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global elementconnectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise elementconnectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. • Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist Ω( log hlog m) elementdisjoint Steiner forests, where h =  i Ti. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving polytime algorithms to find these forests; these are the first nontrivial algorithms for packing elementdisjoint Steiner Forests. • We give a very short and intuitive proof of a spiderdecomposition theorem of Chuzhoy and Khanna [12] in the context of the singlesink kvertexconnectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the elementconnectivity reduction step; we believe it will find more applications in the future.
Approximation Algorithms for Network Design: A Survey
"... In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges ..."
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Cited by 9 (1 self)
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In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges that induces a connected graph (this is the minimumcost spanning tree problem), or we might want to find a minimumcost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimumcost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples. Many practically relevant instances of network design problems are NPhard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomialtime), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an αapproximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the
Approximate MinMax Theorems for Steiner RootedOrientations of Graphs and Hypergraphs
, 2006
"... Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner RootedOrientation problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the “connectivity” from the root r to the vertices in S is maximized. ..."
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Cited by 8 (0 self)
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Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner RootedOrientation problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the “connectivity” from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate minmax relations: • Given an undirected hypergraph H, if S is 2khyperedgeconnected in H, then H has a Steiner rooted khyperarcconnected orientation. • Given an undirected graph G, if S is 2kelementconnected in G, then G has a Steiner rooted kelementconnected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the Steiner Tree Packing problem. Some complementary hardness results are presented at the end.
A primaldual approximation algorithm for the survivable network design problem in hypergraph
 In STACS 2001, number 2010 in Lecture Notes for Computer Science
, 2001
"... Abstract. Given a hypergraph with nonnegative costs on hyperedge and a requirement function r: 2V! Z+, where V is the vertex set, we consider the problem of nding a minimum cost hyperedge set F such that for all S V, F contains at least r(S) hyperedges incident to S. In the case that r is weakly su ..."
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Abstract. Given a hypergraph with nonnegative costs on hyperedge and a requirement function r: 2V! Z+, where V is the vertex set, we consider the problem of nding a minimum cost hyperedge set F such that for all S V, F contains at least r(S) hyperedges incident to S. In the case that r is weakly supermodular (i.e., r(V) = 0 and r(A) + r(B) maxfr(A \B) + r(A [B); r(AB) + r(B A)g for any A;B V), and the socalled minimum violated sets can be computed in polynomial time, we present a primaldual approximation algorithm with performance guarantee dmaxH(rmax), where dmax is the maximum degree of the hyperedges with positive cost, rmax is the maximum requirement, andH(i) = i j=1
Nodeweighted Network Design in Planar and Minorclosed Families of Graphs
"... We consider nodeweighted network design in planar and minorclosed families of graphs. In particular we focus on the edgeconnectivity survivable network design problem (ECSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(uv) for ea ..."
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We consider nodeweighted network design in planar and minorclosed families of graphs. In particular we focus on the edgeconnectivity survivable network design problem (ECSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(uv) for each pair of nodes uv. The goal is to find a minimum nodeweighted subgraph H of G such that, for each pair uv, H contains r(uv) edgedisjoint paths between u and v. Our main result is an O(k)approximation algorithm for ECSNDP where k = maxuv r(uv) is the maximum requirement. This improves the O(k log n)approximation known for nodeweighted ECSNDP in general graphs [15]. Our algorithm and analysis applies to the more general problem of covering a proper function with maximum requirement k. Our result is inspired by, and generalizes, the work of Demaine, Hajiaghayi and Klein [5] who gave constant factor approximation algorithms for nodeweighted Steiner tree and Steiner forest problems (and more generally covering 01 proper functions) in planar and minorclosed families of graphs.
On approximate minmax theorems of graph connectivity problems
, 2006
"... Given an undirected graph G and a subset of vertices S ` V (G), we call the vertices in S the terminal vertices and the vertices in V (G) S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high "connectivity " among the terminal vertices. The ..."
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Given an undirected graph G and a subset of vertices S ` V (G), we call the vertices in S the terminal vertices and the vertices in V (G) S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high &quot;connectivity &quot; among the terminal vertices. The first problem is the Steiner Tree Packing problem, where a Steiner tree is a tree that connects the terminal vertices (Steiner vertices are optional). The goal of this problem is to find a largest collection of edgedisjoint Steiner trees. The second problem is the Steiner RootedOrientation problem. In this problem, there is a root vertex r among the terminal vertices. The goal is to find an orientation of all the edges in G so that the Steiner rootedconnectivity is maximized in the resulting directed graph D. Here, the Steiner rootedconnectivity is defined to be the maximum k so that the root vertex has k arcdisjoint paths to each terminal vertex in D. Both problems are generalizations of two classical graph theoretical problems: the edgedisjoint s, tpaths problem and the edgedisjoint spanning trees problem. Polynomial time algorithms and exact minmax relations are known for the classical problems. However, both problems that we study are NPcomplete, and thus exact minmax relations are not expected. In the following, we say S is ledgeconnected in G if we need to remove at least l edges in order to disconnect two vertices in S. Clearly, the maximum iii l for which S is ledgeconnected in G is an upper bound on the optimal value for both problems that we study (i.e. the number of edgedisjoint Steiner trees, and the Steiner rootedconnectivity in an orientation). The main result of the Steiner Tree Packing problem is the following approximate minmax relation: