On the relationship between the biconnectivity augmentation and travelling salesman problems

Frederickson, Greg N.; JáJá, Joseph F.

doi:10.1016/0304-3975(82)90059-7

Cited by 88 publications

(60 citation statements)

References 15 publications

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“…If β denotes the best known approximation for this problem, the cost of the resulting Nash equilibrium after using the local improvement vector with a start vector from such an algorithm will then be at most 2β times the optimal cost. For k = 2, β = 3/2, from the result of [10], and if the edge lengths satisfy the triangle inequality β = 2 + 2(k−1)/n, from [14]; note that the edge lengths satisfy the triangle inequality only for α = 1, but not for α > 1. When the edge lengths do not satisfy the triangle inequality, no constant factor approximations are known.…”

Section: Proofmentioning

confidence: 99%

Equilibria in Topology Control Games for Ad Hoc Networks

2006

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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 incentive-compatible 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 k-connected, 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 networks, where nodes need to choose neighbors to which they will pay a link. Our work is a first step towards game-theoretic analyses of topology control in wireless and wireline networks.

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Section: Proofmentioning

confidence: 99%

Equilibria in Topology Control Games for Ad Hoc Networks

2006

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“…The following statement is known, and was implicitly proved in [4]. For completeness of exposition, we provide a proof-sketch.…”

Section: Proof Of Lemmamentioning

confidence: 94%

“…Fredrickson and Jájá [4] gave a 2-approximation algorithm for Tree Augmentation, and showed that it is NP-hard even for trees of radius 2 (the radius = ( ) of a tree is ⌈ /2⌉, where is the diameter of ). Cheriyan, Jordán and Ravi [1] proved that Tree Augmentation is NP-hard also in the case of unit costs when is a cycle on the leaves of , and gave a 4/3-approximation algorithm for this version.…”

Section: Tree Augmentationmentioning

confidence: 99%

See 1 more Smart Citation

A (1 + ln 2)-Approximation Algorithm for Minimum-Cost 2-Edge-Connectivity Augmentation of Trees with Constant Radius

Cohen

Nutov

2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. We consider the Tree Augmentation problem: given a graph = ( , ) with edge-costs and a tree on disjoint to , find a minimum-cost edge-subset ⊆ such that ∪ is 2-edge-connected. Tree Augmentation is equivalent to the problem of finding a minimumcost edge-cover ⊆ of a laminar set-family. The best known approximation ratio for Tree Augmentation is 2, even for trees of radius 2. As laminar families play an important role in network design problems, obtaining a better ratio is a major open problem in network design. We give a (1 + ln 2)-approximation algorithm for trees of constant radius. Our algorithm is based on a new decomposition of problem solutions, which may be of independent interest.

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“…On the other hand, lower bounds and heuristics with worst-case guarantees for kECON 1 problems were found for restricted costs, e.g., uniform costs or costs satisfying the triangle inequality, as well as very important results on the structure of optimal survivable networks for this cost structure. Details of these works can be seen in (Bienstock et al, 1990;Cheriyan et al, 2001;Frederickson & Jàjà, 1982;Goemans & Bertsimas, 1993;Goemans & Williamson, 1992;Monma et al, 1990) and in a summarized form in (Stoer, 1992). Unfortunately, there exist few exact algorithms for the NCON and ECON for general costs.…”

Section: Context and Problem Definitionmentioning

confidence: 99%