Multiple cover problem on undirected flow networks

Tamura, Hiroshi; Sugawara, Hidehito; Sengoku, Masakazu; Shoji, Shuichi

doi:10.1002/1520-6440(200101)84:1<67::aid-ecjc7>3.0.co;2-#

Cited by 18 publications

(30 citation statements)

References 4 publications

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“…Location problems in networks are often formulated as optimization problems to determine the best location of facilities such as industrial plants or warehouses in given networks to satisfy a certain property. Location problems based on flow (i.e., connectivity) requirements, called source location problems, were introduced by Tamura et al [17,18], and have recently received much attention from many authors (e.g., [1-3, 10, 11, 14, 20]). …”

Section: Introductionmentioning

confidence: 99%

Minimum Cost Source Location Problems with Flow Requirements

2007

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In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ andκ), where the source location problems are to find a minimum-cost set S ⊆ V in a given graph G = (V , A) with a capacity function u : A → R + such that for each vertex v ∈ V , the connectivity from S to v (resp., from v to S) is at least a given demand d − (v) (resp., d + (v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54-68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of c ln D for some constant c, unless every problem in NP has an O(N log log N )-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1 + ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are (ln v∈V (d + (v) + d − (v)))-approximable.

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Section: Introductionmentioning

confidence: 99%

Minimum Cost Source Location Problems with Flow Requirements

2007

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“…Tamura et al [16][17][18][19] introduced the single source location problem and developped an O(nM (n, m)) algorithm where M (n, m) denotes the time complexity of computing a maximal flow. Initially, the algorithm was only applicable to uniform cost and uniform capacity instances but in the following papers they were able to transfer the algorithm to arbitrary cost and arbitrary capacity instances.…”

Section: Single Source Location Problemmentioning

confidence: 99%

“…Due to the equivalence of uncapacitated single source and single sink problems, we can conclude from Tamura et al [16][17][18][19]: W is an (uncapacitated) sink cover if and only if W ∩ X = ∅ for all minimal deficient sets X of G. W is feasible for the capacitated problem if and only if there exists some w ∈ W ∩ X such that k(W ) ≥ max x∈X a(x).…”

Section: Single Non-simultaneous Independentsink Location Problemmentioning

confidence: 99%

Sink location to find optimal shelters in evacuation planning

Heßler

Hamacher

2016

EURO Journal on Computational Optimization

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The sink location problem is a combination of network flow and location problems: From a given set of nodes in a flow network a minimum cost subset W has to be selected such that given supplies can be transported to the nodes in W . In contrast to its counterpart, the source location problem which has already been studied in the literature, sinks have, in general, a limited capacity. Sink location has a decisive application in evacuation planning, where the supplies correspond to the number of evacuees and the sinks to emergency shelters.We classify sink location problems according to capacities on shelter nodes, simultaneous or non-simultaneous flows, and single or multiple assignments of evacuee groups to shelters. Resulting combinations are interpreted in the evacuation context and analyzed with respect to their worst-case complexity status.There are several approaches to tackle these problems: Generic solution methods for uncapacitated problems are based on source location and modifications of the network. In the capacitated case, for which source location cannot be applied, we suggest alternative approaches which work in the original network. It turns out that latter class algorithms are superior to the former ones. This is established in numerical tests including random data as well as real world data from the city of Kaiserslautern, Germany.

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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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Multiple cover problem on undirected flow networks

Cited by 18 publications

References 4 publications

Minimum Cost Source Location Problems with Flow Requirements

Minimum Cost Source Location Problems with Flow Requirements

Sink location to find optimal shelters in evacuation planning

Transversals of subtree hypergraphs and the source location problem in digraphs

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