Optimal Solution of the Maximum All Request Path Grooming Problem

Bermond, Jean‐Claude; Cosnard, Michel; Coudert, David; Perenn, S.

doi:10.1109/aict-iciw.2006.144

Cited by 10 publications

(6 citation statements)

References 8 publications

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“…Afterwards, DE graphs received attention in papers with applications to optical fibers (see for instance Bermond et al (2007Bermond et al ( , 2006, Caragiannis et al (2001), Erlebach et al (1999)), or from a combinatorial optimization point of view Costa et al (2003Costa et al ( , 2005. The purpose of these studies consists often in the design of polynomial and efficient algorithms to compute various things, such as a minimum proper coloring of I (P, T ) (which can be interpreted as assignement of wavelengths with a minimum number of distinct wavelengths) or a minimum clique cover of I (P, T ) (which is the minimum multicut problem).…”

Section: Introductionmentioning

confidence: 99%

Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

et al. 2010

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International audienceLet T = (V, A) be a directed tree. Given a collection P of dipaths on T, we can look at the arc-intersection graph P whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) and proved that they are perfect. DE graphs find one of their applications in the context of optical networks. For instance, assigning wavelengths to set of dipaths in a directed tree network consists in finding a proper coloring of the arc-intersection graph. In the present paper, we give - a simple algorithm finding a minimum proper coloring of the paths. - a faster algorithm than previously known ones finding a minimum multicut on a directed tree. It runs in O({pipe}V{pipe}{pipe}P{pipe}) (it corresponds to the minimum clique cover of I (P, T)). - a polynomial algorithm computing a kernel in any DE graph whose edges are oriented in a clique-acyclic way. Even if we know by a theorem of Boros and Gurvich that such a kernel exists for any perfect graph, it is in general not known whether there is a polynomial algorithm (polynomial algorithms computing kernels are known only for few classes of perfect graphs)

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Section: Introductionmentioning

confidence: 99%

Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

et al. 2010

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“…Many versions of the problem can be considered, according for example to the routing, the physical graph, and the request graph, among others. For example, in [3, 6] the Path Traffic Grooming problem is studied. If the network topology is a ring (which is the case of SONET rings), we mainly distinguish two cases depending on the routing.…”

Section: Introductionmentioning

confidence: 99%

Traffic grooming in bidirectional WDM ring networks

Bermond

Muñoz

2010

Networks

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International audienceWe study the minimization of ADMs (Add-Drop Multiplexers) in optical WDM bidirectional rings considering symmetric shortest path routing and all-to-all unitary requests. We precisely formulate the problem in terms of graph decompositions, and state a general lower bound for all the values of the grooming factor C and N, the size of the ring. We first study exhaustively the cases C = 1, C = 2, and C = 3, providing improved lower bounds, optimal constructions for several infinite families, as well as asymptotically optimal constructions and approximations. We then study the case C > 3, focusing specifically on the case C = k(k + 1)/2 for some k ≥ 1. We give optimal decompositions for several congruence classes of N using the existence of some combinatorial designs. We conclude with a comparison of the cost functions in unidirectional and bidirectional WDM rings

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“…A coloring of I(P, T ) is a color assignment to the dipaths is such a way that two dipaths sharing a common arc get distinct colors. The color assignment problem is often interpreted as a wavelength assignment on an optical fiber (see [3,2] for further studies from this point of view). A clique cover of I(P, T ) is a set of arcs intersecting all dipaths.…”

Section: Introductionmentioning

confidence: 99%

Directed paths on a tree: coloring, multicut and kernel

de Gévigney,

Meunier,

Popa

et al. 2009

Preprint

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In a paper published in Journal of Combinatorial Theory, Series B (1986), Monma and Wei propose an extensive study of the intersection graph of paths on a tree. They explore this notion by varying the notion of intersection: the paths are respectively considered to be the sets of their vertices and the sets of their edges, and the trees may or may not be directed. Their main results are a characterization of these graphs in term of their clique tree and a unified recognition algorithm. Related optimization problems are also presented.In the present note, we are interested by the case when the tree is directed, and the paths are considered to be the set of their arcs. An application is the question of wavelength assignment in optimal network. In their article, Monma and Wei show that in this case the intersection graph is a perfect graph and they give polynomial combinatorial algorithms to solve the minimum coloring and the maximum stable problems, with the help of the clique tree representation (Tarjan's method). The maximum clique problem is immediate. They leave the problem of finding a minimum clique cover, which is here nothing else than the minimum multicut of the dipaths, as an open question. Costa, Létocart and Roupin in a survey about multicut and integer multiflow noted that the maximum stable, which is a maximum set of arc-disjoint dipaths, and the minimum clique cover, which is the minimum set of arcs intersecting all dipaths (minimum multicut), are integers solutions of a linear program with totally unimodular matrices, and hence can be solved polynomially. For the maximum stable problem (or maximum set of arc-disjoint dipaths), there is a simple polynomial algorithm found by Garg, Vazirani and Yannakakis.In the present paper, we present faster algorithms for solving the minimum coloring and the minimum clique cover problems (minimum multicut), and maybe simpler, since we work directly with the dipaths on the tree and avoid both the clique decomposition and the linear programming formulation. They both run in O(np) time, where n is the number of vertices of the tree and p the number of paths. Another result is a polynomial algorithm computing a kernel in the intersection graph, when its edges are oriented in a clique-acyclic way. Indeed, such a kernel exists for any perfect graph by a theorem of Boros and Gurvich. Such algorithms computing kernels are known only for few classes of perfect graphs.

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Optimal Solution of the Maximum All Request Path Grooming Problem

Cited by 10 publications

References 8 publications

Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

Traffic grooming in bidirectional WDM ring networks

Directed paths on a tree: coloring, multicut and kernel

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