Gallo, G., G. Longo, S. Pallottino, and S. Nguyen (1993). Directed hypergraphs and applications. Discrete Applied Mathematics 42, 177--201.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....basic definitions of hypergraphs, in particular of directed hypergraphs; in Subsection 5.2 we use these concepts to give a topological description of the three levels of an agency. 5. 1 Directed Hypergraph Hypergraphs were studied by Berge [1] 2] 3] some of their applications are presented in [17]. Here we give only the definitions useful for representing, from a topological point of view, the levels of an agency. Definition 5.1.1: an hypergraph is a pair H= N, E) where N= n 1 , n 2 , n n is the set of nodes and E= E 1 , E 2 , Em is the set of hyperedges, with E i N for i=1, ....

G. Gallo, G. Longo, S. Pallottino, S. Nguyen, "Directed hypergraphs and applications", Discrete Applied Mathematics, 42, 1993, p. 177-201

....and co partitions. 2 Preliminaries on directed hypergraphs The concept of directed hypergraphs was introduced in many di erent contexts, in areas like propositional logic, assembly, and relational databases, to eciently model many to one relations; surveys of these applications can be found in [6] and [7] In our terminology, a directed hypergraph is a pair H = V; E) where V is a nite ground set, and E is a nite collection of so called hyperarcs (possibly with repetition) a hyperarc is a subset Z V with a designated head node v 2 Z, and it is denoted by Z v . The nodes ....

G. Gallo, G. Longo, S. Nguyen and S. Pallottino, Directed hypergraphs and applications, Discrete Appl. Math. 40 (1993) 177-201.

....stop. For more details, see [Nguyen Pallottino, 1986, 1988, 1989; Marcotte Nguyen, 1997] The transit networks introduced above can be modeled as particular directed hypergraphs, a generalization of the classical directed graphs (for more details about the theory of hypergraphs and hyperpaths, see [Gallo, Longo, Nguyen Pallottino, 1992]) In the case of equilibrium transit assignment, it is possible to formulate the problem in terms of hypergraphs, and based on this formulation, solve the problem by iteratively computing shortest hyperpaths . As shown in [Gallo, Longo, Nguyen Pallottino, 1992] classical shortest path ....

....of hypergraphs and hyperpaths, see [Gallo, Longo, Nguyen Pallottino, 1992] In the case of equilibrium transit assignment, it is possible to formulate the problem in terms of hypergraphs, and based on this formulation, solve the problem by iteratively computing shortest hyperpaths . As shown in [Gallo, Longo, Nguyen Pallottino, 1992], classical shortest path algorithms can be generalized to find certain classes of shortest hyperpaths in directed hypergraphs, with a low computational time complexity. In particular, some shortest hyperpath algorithms have been adapted to the particular hypergraph structure of the transit ....

G. Gallo, G. Longo, S. Nguyen and S. Pallottino (1992), "Directed hypergraphs and applications", Discrete Applied Mathematics 40, 177-201.

.... V , for 4 1 2 3 4 5 6 e 1 e 2 e 3 e 4 1 2 3 4 5 6 e 1 e 4 e 3 e 2 R Figure 1: A hyperpath and one of its permutations which R y (connectivity procedure) It is easy to see that R is not connected to y, i.e. R 6 y, if and only if a Ry cut of cardinality 0 exists in H [14]. Assume that each hyperarc e in H is assigned a real cost c(e) Given a hyperpath P Rt from R to t, by cost function we mean a node function C which assigns a cost to each node in P Rt depending on the costs of its hyperarcs. C(t) is the cost of P Rt under the chosen cost function. Additive ....

.... Equations (3) More precisely, algorithms have been proposed which run in O(n Theta size(H) time, in the general case, and in O(maxfn 2 ; size(H)g) time or in O(maxfmlogn; size(H)g) time, if c(e) F (fC(w) w 2 T e g) C(w) for each w 2 T e and for each e 2 E (Dijkstra s like condition) [14]. Observe that this last condition holds for additive cost functions such that F (fC(w) w 2 T e g) MaxfC(w) w 2 T e g, which is the case considered by Knuth in [27] In the simple case in which the hypergraph is acyclic, the minimum cost hyperpaths rooted at R can be computed in O(size(H) ....

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G. Gallo, G. Longo, S. Nguyen, and S. Pallottino. Directed hypergraphs and applications. Discrete Appl. Math., 40:177--201, 1993.

.... [2,4] A recent, quite promising approach is based on the use of hypergraph models to represent and derive assembly plans [2,4,8] Directed hypergraphs have proven to be an elegant and computationally efficient tool to model and to solve several relevant combinatorial optimization problems [1, 5, 6, 7, 10, 11, 12]. In [5, 6] an attempt is made to lay down the basic blocks for a theory of directed hypergraphs. Here we consider the assembly problem as defined in [8] together with some variants, and present computationally efficient algorithms for its solution. The paper is organized as follows. In the ....

.... promising approach is based on the use of hypergraph models to represent and derive assembly plans [2,4,8] Directed hypergraphs have proven to be an elegant and computationally efficient tool to model and to solve several relevant combinatorial optimization problems [1, 5, 6, 7, 10, 11, 12] In [5, 6] an attempt is made to lay down the basic blocks for a theory of directed hypergraphs. Here we consider the assembly problem as defined in [8] together with some variants, and present computationally efficient algorithms for its solution. The paper is organized as follows. In the second ....

[Article contains additional citation context not shown here]

Gallo, G., G. Longo, S. Nguyen and S. Pallottino: "Directed hypergraphs and applications", Techn.Rept. TR3 /90, Dipartimento di Informatica, Universit di Pisa (1990) forthcoming in Discrete Applied Mathematics.

.... C s k )g 8j 2 N Gamma frg: Assuming that the combinatorial problem of determining the optimal set E s j : E s j = arg min E s j 2E j fw s j X k2E s j s jk (c jk C s k )g is well defined, the following generic shortest hyperpath algorithm, similar to the one given in Gallo et al. 1993), produces a shortest hyperpath tree with root r: HYPERPATH FORMULATIONS 5 ALGORITHM HYPERSHORT input: destination r output: fE s j ; C s j g j2N INITIALIZATION for j 2 N do C s j : 1 E s j : endfor C s r : 0 Q : N Gamma frg for j 2 r Gamma do E s j : ....

....suggested. 1.2. 4 The oriented hypergraph framework The hyperpath concept induced by the strategic modeling of passenger or vehicle assignment on transport networks may be embbeded within a more classical framework, namely that of Berge s hypergraph (Berge, 1973) Using the notation introduced in Gallo et al. 1993), we define a directed hypergraph as a pair H = V ; E) where V is the set of nodes, and E is the set of hyperarcs. A hyperarc e 2 E is a pair (e Gamma ; e ) where e Gamma and e , respectively the tail and the head of e, are non empty disjoint subsets of V . Clearly, each pair (j; ....

Gallo, G., Longo, G., Nguyen, S., and Pallottino, S. (1993). Directed hypergraphs and applications. Discrete Applied Mathematics, 42:177--201.

.... the desired product have been proposed in the literature and used in practice (De Fazio and Whitney, 1987; Baldwin, Abell, Lui, De Fazio and Whitney, 1991) Recently, the use of hypergraph models has been proposed to represent and derive assembly plans (Homem De Mello and Sanderson, 1990; Gallo and Pallottino, 1992). In particular, in [Gallo and Pallottino, 1992] the problem of finding optimal assembly plans with unbounded parallelism has been formulated and solved in terms of Shortest Hyperpath computations. Moreover, a heuristics has been described for the problem of assigning the operations of a given ....

.... in the literature and used in practice (De Fazio and Whitney, 1987; Baldwin, Abell, Lui, De Fazio and Whitney, 1991) Recently, the use of hypergraph models has been proposed to represent and derive assembly plans (Homem De Mello and Sanderson, 1990; Gallo and Pallottino, 1992) In particular, in [Gallo and Pallottino, 1992] the problem of finding optimal assembly plans with unbounded parallelism has been formulated and solved in terms of Shortest Hyperpath computations. Moreover, a heuristics has been described for the problem of assigning the operations of a given assembly plan to the machines in the case of ....

[Article contains additional citation context not shown here]

GALLO, G. , LONGO, G. , NGUYEN, S. , AND S. PALLOTTINO. 1992. Directed hypergraphs and applications. Discrete Applied Mathematics 40, 177-201.

....of minimum cost need to be computed several times in the context under consideration. The transit networks introduced above can be modeled as particular directed hypergraphs, a generalization of classical directed graphs (for more details about the theory of hypergraphs and hyperpaths, see [Gallo, Longo, Nguyen and Pallottino, 1992]) In the case of equilibrium transit assignment, it is possible to solve the problem by iteratively computing shortest hyperpaths with respect to the expected costs . As shown in [Gallo, Longo, Nguyen and Pallottino, 1992] classical shortest path algorithms can be generalized to find 40 ....

....(for more details about the theory of hypergraphs and hyperpaths, see [Gallo, Longo, Nguyen and Pallottino, 1992] In the case of equilibrium transit assignment, it is possible to solve the problem by iteratively computing shortest hyperpaths with respect to the expected costs . As shown in [Gallo, Longo, Nguyen and Pallottino, 1992], classical shortest path algorithms can be generalized to find 40 certain classes of shortest hyperpaths in directed hypergraphs, with a low computational time complexity. In particular, some shortest hyperpath algorithms have been adapted to the particular hypergraph structure of the transit ....

G. Gallo, G. Longo, S. Nguyen and S. Pallottino (1992). "Directed hypergraphs and applications". Discrete Applied Mathematics 40, 177-201.

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Gallo, G., G. Longo, S. Pallottino, and S. Nguyen (1993). Directed hypergraphs and applications. Discrete Applied Mathematics 42, 177--201.

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Gallo, G., Longo, G., and Pallottino, S. (1993). Directed hypergraphs and applications. Discrete Applied Mathematics, 42(2), 177--201.

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Gallo, G., Longo, G., Pallottino, S., and Nguyen, S., "Directed hypergraphs and applications," Discrete Applied Mathematics 42 pp. 177-201 (1993).

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Gallo, G., Longo, G., Pallottino, S., and Nguyen, S., "Directed hypergraphs and applications," Discrete Applied Mathematics 42 pp. 177-201 (1993).

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