Results 11  20
of
79
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 ..."
Abstract

Cited by 50 (4 self)
 Add to MetaCart
(Show Context)
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.
Equilibria in topology control games for ad hoc networks
 in: Proceedings in DIALMPOMC Mobicom
, 2003
"... Abstract. We study topology control problems in ad hoc networks where network nodes get to choose their power levels in order to ensure desired connectivity properties. Unlike most other work on this topic, we assume that the network nodes are owned by different entities, whose only goal is to maxim ..."
Abstract

Cited by 47 (5 self)
 Add to MetaCart
(Show Context)
Abstract. We study topology control problems in ad hoc networks where network nodes get to choose their power levels in order to ensure desired connectivity properties. Unlike most other work on this topic, we assume that the network nodes are owned by different entities, whose only goal is to maximize their own utility that they get out of the network without considering the overall performance of the network. Game theory is the appropriate tool to study such selfish nodes: we define several topology control games in which the nodes need to choose power levels in order to connect to other nodes in the network to reach their communication partners while at the same time minimizing their costs. We study Nash equilibria and show that—among the games we define—these can only be guaranteed to exist if each network node is required to be connected to all other nodes (we call this the STRONG CONNECTIVITY GAME). For a variation called CONNECTIVITY GAME, where each node is only required to be connected (possibly via intermediate nodes) to a given set of nodes, we show that Nash equilibria do not necessarily exist. We further study how to find Nash equilibria with incentivecompatible algorithms and compare the cost of Nash equilibria to the cost of a social optimum, which is a radius assignment that minimizes the total cost in a network where nodes cooperate. We also study variations of the games; one where nodes not only have to be connected, but kconnected, and one that we call the REACHABILITY GAME, where nodes have to reach as many other nodes as possible, while keeping costs low. We extend our study of the STRONG CONNECTIVITY GAME and the CONNECTIVITY GAME to wireless networks with directional antennas and wireline
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) ..."
Abstract

Cited by 44 (0 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 43 (3 self)
 Add to MetaCart
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].
Relay Placement for Higher Order Connectivity in Wireless Sensor Networks
"... Sensors typically use wireless transmitters to communicate with each other. However, sensors may be located in a way that they cannot even form a connected network (e.g, due to failures of some sensors, or loss of battery power). In this paper we consider the problem of adding the smallest number o ..."
Abstract

Cited by 40 (2 self)
 Add to MetaCart
Sensors typically use wireless transmitters to communicate with each other. However, sensors may be located in a way that they cannot even form a connected network (e.g, due to failures of some sensors, or loss of battery power). In this paper we consider the problem of adding the smallest number of additional (relay) nodes so that the induced communication graph is 2connected 1. The problem is NPhard. In this paper we develop O(1)approximation algorithms that find close to optimal solutions in time O((kn) 2) for achieving kedge connectivity of n nodes. The worst case approximation guarantee is 10, but the algorithm produces solutions that are far better than this bound suggests. We also consider extensions to higher dimensions, and the scheme that we develop for points in the plane, yields a bound of 2dMST where dMST is the maximum degree of a minimumdegree Minimum Spanning Tree in d dimensions using Euclidean metrics. In addition, our methods extend with the same approximation guarantees to a generalization when the locations of relays are required to avoid certain polygonal regions (obstacles). We also prove that if the sensors are uniformly and identically distributed in a unit square, the expected number of relay nodes required goes to zero as the number of sensors goes to infinity.
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 ..."
Abstract

Cited by 35 (13 self)
 Add to MetaCart
(Show Context)
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).
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 ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
(Show Context)
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.
Algorithms for SingleSource Vertex Connectivity
"... In the Survivable Network Design Problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithm ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
(Show Context)
In the Survivable Network Design Problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithmic techniques. For the edgeconnectivity version of the problem, a 2approximation algorithm is known for arbitrary pairwise connectivity requirements. However, no nontrivial algorithms are known for its vertex connectivity counterpart. In fact, even highly restricted special cases of the vertex connectivity version remain poorly understood. We study the singlesource kvertex connectivity version of SNDP. We are given a graph G(V, E) with a subset T of terminals and a source vertex s, and the goal is to find a minimum cost subset of edges ensuring that every terminal is kvertex connected to s. Our main result is an O(k log n)approximation algorithm for this problem; this improves upon the recent 2 O(k2) log 4 napproximation. Our algorithm is based on an intuitive rerouting scheme. The analysis relies on a structural result that may be of independent interest: we show that any solution can be decomposed into a disjoint collection of multiplelegged spiders, which are then used to reroute flow from terminals to the source via other terminals. We also obtain the first nontrivial approximation algorithm for the vertexcost version of the same problem, achieving an O(k 7 log 2 n)approximation. 1.
Network Design for Vertex Connectivity
 In Proceedings of ACM Symposium on Theory of Computing (STOC), 2008. 6 C. Chekuri and
, 2008
"... We study the survivable network design problem (SNDP) for vertex connectivity. Given a graph G(V, E) with costs on edges, the goal of SNDP is to find a minimum cost subset of edges that ensures a given set of pairwise vertex connectivity requirements. When all connectivity requirements are between a ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
(Show Context)
We study the survivable network design problem (SNDP) for vertex connectivity. Given a graph G(V, E) with costs on edges, the goal of SNDP is to find a minimum cost subset of edges that ensures a given set of pairwise vertex connectivity requirements. When all connectivity requirements are between a special vertex, called the source, and vertices in a subset T ⊆ V, called terminals, the problem is called the singlesource SNDP. Our main result is a randomized k O(k2) log 4 napproximation algorithm for singlesource SNDP where k denotes the largest connectivity requirement for any sourceterminal pair. In particular, we get a polylogarithmic approximation for any constant k. Prior to our work, no nontrivial approximation guarantees were known for this problem for any k ≥ 3. We also show that SNDP is k Ω(1)hard to approximate and provide an elementary construction that shows that the wellstudied setpair linear programming relaxation for this problem has an ˜ Ω(k 1/3) integrality gap.