Minimum cost source location problem with vertex-connectivity requirements in digraphs

Nagamochi, Hiroshi; Ishii, Toshimasa; Ito, Hiro

doi:10.1016/s0020-0190(01)00183-1

Cited by 33 publications

(26 citation statements)

References 7 publications

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“…A polynomial-time algorithm for this problem is given in a paper by Nagamochi et al [12]. They show that the problem has a matroidal property, as follows.…”

Section: Optimally Deciding Which Nodes To Verifymentioning

confidence: 99%

False-Name-Proofness in Social Networks

Conitzer

Immorlica²,

Letchford

et al. 2010

Lecture Notes in Computer Science

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In mechanism design, the goal is to create rules for making a decision based on the preferences of multiple parties (agents), while taking into account that agents may behave strategically. An emerging phenomenon is to run such mechanisms on a social network; for example, Facebook recently allowed its users to vote on its future terms of use. One significant complication for such mechanisms is that it may be possible for a user to participate multiple times by creating multiple identities. Prior work has investigated the design of false-name-proof mechanisms, which guarantee that there is no incentive to use additional identifiers. Arguably, this work has produced mostly negative results. In this paper, we show that it is in fact possible to create good mechanisms that are robust to false-name-manipulation, by taking the social network structure into account. The basic idea is to exclude agents that are separated from trusted nodes by small vertex cuts. We provide key results on the correctness, optimality, and computational tractability of this approach.

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“…A polynomial-time algorithm for this problem is given in a paper by Nagamochi et al [12]. They show that the problem has a matroidal property, as follows.…”

Section: Optimally Deciding Which Nodes To Verifymentioning

confidence: 99%

False-Name-Proofness in Social Networks

Conitzer

Immorlica²,

Letchford

et al. 2010

Lecture Notes in Computer Science

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“…( For other examples of location problems with connectivity requirements see [1,6,7,8,9,10,12,13] ). Ito et al [8] found an algorithm for the Source Location Problem that was polynomial for fixed k and l. They left open the problem of finding an algorithm that is polynomial when k and l are not fixed.…”

Section: Constraint : λ(S V) ≥ K and λ(V S) ≥ L For All V ∈ V \ Smentioning

confidence: 99%

Transversals of subtree hypergraphs and the source location problem in digraphs

Heuvel

Johnson

2007

Networks

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A hypergraph H = (V, E) is a subtree hypergraph if there is a tree T on V such that each hyperedge of E induces a subtree of T . To find a minimum size transversal for a subtree hypergraph is, in general, NP-hard. In this paper, we show that if it is possible to decide if a set of vertices W ⊆ V is a transversal in time S(n) ( where n = |V | ), then it is possible to find a minimum size transversal in O(n 3 S(n)). This result provides a polynomial algorithm for the Source Location Problem : a set of (k, l)-sources for a digraph D = (V, A) is a subset K of V such that for any v ∈ V \ K there are k arc-disjoint paths that each join a vertex of K to v and l arc-disjoint paths that each join v to K. The Source Location Problem is to find a minimum size set of (k, l)-sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case S(n) is polynomial.

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“…We cannot expect that, in general, the Source Location Problem ( which has been extensively studied ) will provide solutions to the External Network Problem : the Source Location Problem with vertex-connectivity requirements ( see [3,5] ) is not equivalent to the natural formulation of the External Network Problem with vertex-connectivity requirements. This is discussed further in Section 8.…”

Section: External Network Problemmentioning

confidence: 99%

The External Network Problem with Edge- or Arc-Connectivity Requirements

Heuvel

Johnson

2005

Combinatorial and Algorithmic Aspects of Networking

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van den Heuvel, J. and Johnson, M. (2005) 'The external network problem with edge-or arc-connectivity requirements.', in Combinatorial and algorithmic aspects of networking : rst workshop on combinatorial and algorithmic aspects The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract The connectivity of a communications network can often be enhanced if the nodes are able, at some expense, to form links using an external network. In this paper, we consider the problem of how to obtain a prescribed level of connectivity with a minimum number of nodes connecting to the external network. Let D = (V, A) be a digraph. A subset X of vertices in V may be chosen, the so-called external vertices. An internal path is a normal directed path in D; an external path is a pair of internal paths p 1 = v 1 · · · v s , p 2 = w 1 · · · w t in D such that v s and w 1 are external vertices (the idea is that v 1 can contact w t along this path using an external link from v s to w 1). Then D is externally-k-arc-strong if for each pair of vertices u and v in V , there are k arc-disjoint paths (which may be internal or external) from u to v. We present polynomial algorithms that, given a digraph D and positive integer k, will find a set of external vertices X of minimum size subject to the requirement that D must be externally-k-arc-strong. We also consider two generalisations of the problem : first we suppose that the number of arc-disjoint paths required from u to v may differ for each choice of u and v, and secondly we suppose that each vertex has a cost and that we are required to find a minimum cost set of external vertices. We show that these two problems are NP-hard. Finally, we consider the analogue of this problem for vertex-connectivity in undirected graphs. A graph G with set of external vertices X is externally-k-connected if there are k vertex-disjoint paths joining each pair of vertices in G. We present polynomial algorithms for finding a minimum size set of external vertices subject to the requirement that G must be externally-k-connected for the cases k ∈ {1, 2, 3}.

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Minimum cost source location problem with vertex-connectivity requirements in digraphs

Cited by 33 publications

References 7 publications

False-Name-Proofness in Social Networks

False-Name-Proofness in Social Networks

Transversals of subtree hypergraphs and the source location problem in digraphs

The External Network Problem with Edge- or Arc-Connectivity Requirements

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