Algorithms for central-median paths with bounded length on trees

Becker, Ronald I.; Lari, Isabella; Scozzari, Andrea

doi:10.1016/j.ejor.2005.09.049

Cited by 20 publications

(17 citation statements)

References 28 publications

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“…A linear time algorithm was developed for the following three problems: (1) the minimization of a convex combination of the above pair of criteria (the cent-dian problem), (2) the minimization of the weighted median criterion subject to an upper bound on the maximum unweighted distance from the selected path, and (3) the minimization of the unweighted maximum distance subject to an upper bound on the median objective. Becker et al [4,5] study the generalization of the last two problems to the case where the length of the selected path cannot exceed a prespecified value L. They solve this pair of constrained minimization problems in O(n log 2 n) time. In this article we extend, generalize, and improve upon the results in [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

“…Becker et al [4,5] study the generalization of the last two problems to the case where the length of the selected path cannot exceed a prespecified value L. They solve this pair of constrained minimization problems in O(n log 2 n) time. In this article we extend, generalize, and improve upon the results in [3][4][5]. We also consider tree graphs only, and use a bi-criteria model to represent the total transportation cost of the customers (nodes) to the facility (path).…”

Section: Introductionmentioning

confidence: 99%

“…We note that in the models in [3][4][5], all the center weights are identical. (This uniformity assumption simplifies the combinatorial complexity of the model significantly.…”

Section: Introductionmentioning

confidence: 99%

“…If F can be evaluated at each planar point in constant time, we conclude that a path minimizing F(MAX(P), SUM(P)) can be obtained in O(n) time by evaluating F at each point of W L . For example, we can solve the minimization problems mentioned below [(7)-(9)] in linear time by enumerating all the points in W L , improving and extending the results in [3][4][5]. When the set NDO(P L ) is given and ordered according to the MAX coordinates, we can generate the extreme points of its convex hull in O(n) time, and then solve each of these minimization problems in O(log n) time.…”

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confidence: 98%

“…Note that articles [4,5] consider only the version where the center criterion is unweighted and give O(n log 2 n) algorithms for solving the last pair of problems. We solve these problems even for the doubly weighted model in O(n log n) time.…”

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confidence: 99%

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The bi‐criteria doubly weighted center‐median path problem on a tree

Puerto

Rodríguez‐Chía

Tamir³

et al. 2006

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Given a tree network T with n nodes, let PL be the subset of all discrete paths whose length is bounded above by a prespeciﬁed value L. We consider the location of a path-shaped facility P ∈ PL, where customers are represented by the nodes of the tree. We use a bi-criteria model to represent the total transportation cost of the customers to the facility. Each node is associated with a pair of nonnegative weights: the center-weightand the median-weight. In this doubly weighted model, a path P is assigned a pair of values (MAX (P ), SUM (P )),which are, respectively, the maximum center-weighted distance and the sum of the median-weighted distances from P to the nodes of the tree. Viewing PL and the planar set {(MAX (P), SUM (P )) : P ∈ PL} as the decisionspace and the bi-criteria or outcome space respectively,we focus on ﬁnding all the nondominated points of the bi-criteria space. We prove that there are at most 2n non-dominated outcomes, even though the total number of efﬁcient paths can be Ω(n2), and they can all be generated in O(n log n) optimal time. We apply this result to solve the cent-dian model, whose objective is a convex combination of the weighted center and weighted median functions. We also solve the restricted models, where the goal is to minimize one of the two functions MAX or SUM, subject to an upper bound on the other one, both withand without a constraint on the length of the path. All these problems are solved in linear time, once the set of non dominated outcomes has been obtained, which in turn, results in an overall complexity of O(n log n). The latter bounds improve upon the best known results by a factor of O(log n).Ministerio de Ciencia y Tecnologí

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We note that in the models in [3][4][5], all the center weights are identical. (This uniformity assumption simplifies the combinatorial complexity of the model significantly.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

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The bi‐criteria doubly weighted center‐median path problem on a tree

Puerto

Rodríguez‐Chía

Tamir³

et al. 2006

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The continuous and discrete path‐variance problems on trees

2008

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In this article we consider the problem of locating pathshaped facilities on a tree network, minimizing the variance objective function. This type of objective is generally adopted in location problems arising in public sector applications, such as the location of evacuation or mass transit routes. We consider a weighted tree, in which a positive weight is assigned to each vertex of the tree, and positive real lengths are associated with its edges. We study the general case in which the path is continuous, that is, the end points of the optimal path can be either vertices, or points along an edge, and there is an upper bound on the length of the path. Given a tree with n vertices, for this problem we provide an O(n 2) algorithm, and we show how it can be applied, with the same complexity, to the discrete case, that is, when the end points of the optimal path are vertices of the tree. We improve the previous best complexity bound in (Cáceres et al., Discr Appl Math 145 (2004), 72-79), for the unrestricted length continuous path-variance problem, by a factor of log n. We also show that the optimal point for the variance objective function does not satisfy any nestedness property with respect to the optimal path in the unconstrained (discrete or continuous) version of the problem. © 2008 Wiley Periodicals, Inc

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Minimax regret path location on trees

Puerto

Ricca

Scozzari³

2011

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This work studies the problem of finding optimal paths with respect to the center, median and centdian objective functions, on networks with uncertain vertex weights that are given as intervals. Our approach looks for minimax regret paths which minimize the worst-case opportunity loss in the corresponding objective function. These problems are NP-hard on general graphs, therefore we study them on trees. We show that a discrete optimal path always exists for each of them, and provide polynomial time solution algorithms.

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Algorithms for central-median paths with bounded length on trees

Cited by 20 publications

References 28 publications

The bi‐criteria doubly weighted center‐median path problem on a tree

The bi‐criteria doubly weighted center‐median path problem on a tree

The continuous and discrete path‐variance problems on trees

Minimax regret path location on trees

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