An Algorithmic Approach to Network Location Problems. I: The<i>p</i>-Centers

Kariv, Oded; Hakimi, S. L.

doi:10.1137/0137040

Cited by 593 publications

(354 citation statements)

References 17 publications

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“…The natural question that arises is whether the same problem is polynomially solvable in simpler graph topologies. For some well-known problems in location analysis, as the p-center or p-median [8,9], this is the case when the model is restricted to tree networks. Thus, the main goal in this paper is to answer whether MCLP is polynomially solvable on trees.…”

Section: Introductionmentioning

confidence: 99%

Polynomial algorithms for partitioning a tree into single‐center subtrees to minimize flat service costs

et al. 2007

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This paper deals with the following graph partitioning problem: given a graph with n nodes, p of which are prescribed to be centers (the remaining nodes are called units), the goal is to find a partition of the set of nodes into connected components containing only one center each, so as to minimize the total assignment cost of units to centers. This problem is known to be NP-hard in general graphs, and it is shown here to remain such even if the assignment cost is monotone and the graph is bipartite. Therefore, we specialize on tree graphs to derive polynomial time algorithms. For this class of graphs we provide several reformulations of the problem as integer linear programs. Moreover, we develop a dynamic programming algorithm, whose recursion is based on solving sequences of minimum weight closure problems, that solves the problem on trees in O(n 2 p).

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Section: Introductionmentioning

confidence: 99%

Polynomial algorithms for partitioning a tree into single‐center subtrees to minimize flat service costs

et al. 2007

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“…This latter definition by the sum, if divided by n, is regarded as the average distance from the set to all vertices, and thus called the average k-center problem. These two problems are defined in Kariv and Hakimi [11], [12] and Gary and Johnson [8], and shown to be NP-complete. When k = 2, straightforward algorithms of O(n 3 ) time are known for the two 2-center problems.…”

Section: Introductionmentioning

confidence: 99%

“…When k = 2, straightforward algorithms of O(n 3 ) time are known for the two 2-center problems. In [11] and [12] O(n 2 log n) time and O(n 2 ) time algorithm for the absolute and average k-center problems are shown for trees. Frederickson [6] gives a linear time algorithm for the k-center problems for a tree with unit edge costs.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for the 2-Center Problems

Takaoka

2010

Computational Science and Its Applications – ICCSA 2010

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Abstract. This paper achieves O(n 3 log log n/ log n) time for the 2-center problems on a directed graph with non-negative edge costs under the conventional RAM model where only arithmetic operations, branching operations, and random accessibility with O(log n) bits are allowed. Here n is the number of vertices. This is a slight improvement on the best known complexity of those problems, which is O(n 3 ). We further show that when the graph is with unit edge costs, one of the 2-center problems can be solved in O(n 2.575 ) time.

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“…In [3], Hassin and Tamir showed that the graph-theoretic version of the minimum diameter spanning tree (MDST) problem, where the edge costs do not necessarily satisfy the triangle inequality, is reducible to the absolute 1-center problem introduced by Hakimi [2]. The absolute 1-center problem can be solved in O(mn + n 2 logn) time [6], where m and n denote respectively the numbers of edges and of vertices of G. Chan [10] improved the bound of GMDST problem in d-dimensional space R d . He described a semi-online model that computes GMDST of an n-point set P ⊂ R d withinÕ(n 3− 1 (d+1)(d/2+1) ) time by maintaining a dynamic data structure.…”

Section: Introductionmentioning

confidence: 99%

Geometric Minimum Diameter Minimum Cost Spanning Tree Problem

Seo

Lee

Lin

2009

Algorithms and Computation

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Abstract. In this paper we consider bi-criteria geometric optimization problems, in particular, the minimum diameter minimum cost spanning tree problem and the minimum radius minimum cost spanning tree problem for a set of points in the plane. The former problem is to construct a minimum diameter spanning tree among all possible minimum cost spanning trees, while the latter is to construct a minimum radius spanning tree among all possible minimum cost spanning trees. The graphtheoretic minimum diameter minimum cost spanning tree (MDMCST) problem and the minimum radius minimum cost spanning tree (MRM-CST) problem have been shown to be NP-hard. We will show that the geometric version of these two problems, GMDMCST problem and GMRMCST problem are also NP-hard. We also give two heuristic algorithms, one MCST-based and the other MDST-based for the GMDMCST problem and present some experimental results.

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An Algorithmic Approach to Network Location Problems. I: Thep-Centers

Cited by 593 publications

References 17 publications

Polynomial algorithms for partitioning a tree into single‐center subtrees to minimize flat service costs

Polynomial algorithms for partitioning a tree into single‐center subtrees to minimize flat service costs

Efficient Algorithms for the 2-Center Problems

Geometric Minimum Diameter Minimum Cost Spanning Tree Problem

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