Abstract
Let T = (V, A) be a directed tree. Given a collection \({\mathcal{P}}\) of dipaths on T, we can look at the arc-intersection graph \({I(\mathcal{P},T)}\) whose vertex set is \({\mathcal{P}}\) 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
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a simple algorithm finding a minimum proper coloring of the paths.
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a faster algorithm than previously known ones finding a minimum multicut on a directed tree. It runs in \({O(|V||\mathcal{P}|)}\) (it corresponds to the minimum clique cover of \({I(\mathcal{P},T)}\)).
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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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References
Aharoni R, Holzman R (1998) Fractional kernels in digraphs. J Comb Theor (ser B) 73: 1–6
Bermond J-C, Braud L, Coudert D (2007) Traffic grooming on the path. Theor Comput Sci 384: 139–151
Bermond J-C, Cosnard M, Coudert D, Pérennes S (February 2006) Optimal solution of the maximum all request path grooming problem. In: Proceedings of the advanced international conference on telecommunications, AICT06, Guadeloupe, France
Boros E, Gurvich V (1996) Perfect graphs are perfect solvable. Discret Math 159: 35–55
Caragiannis I, Kaklamanis C, Persiano P (2001) Wavelength routing in all-optical tree networks: A survey. Comput Artif Intell 20: 95–120
Costa M-C, Létocart L, Roupin F (2003) A greedy algorithm for multicut and integral multiflow in rooted trees. Oper Res Lett 31: 21–27
Costa M-C, Létocart L, Roupin F (2005) Minimal multicut and maximal integer multiflow: a survey. Eur J Oper Res 162: 55–69
Erlebach T, Munchen T, Jansen K, Kaklamanis C, Persiano P (1999) Optimal wavelength routing on directed fiber trees. Theor Comput Sci 221: 119–137
Gale D, Shapley, LS (1962) College admissions and the stability of marriage. American Mathematical Monthly, pp 9–15
Garg N, Vazirani VV, Yannakakis Md (1997) Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18: 3–20
Grötschel M, Lovász L, Schrijver A (1988) Geometric algorithms and combinatorial optimization. Springer, Berlin
Lovász L (1972) Normal hypergraphs and the perfect graph conjecture. Discret Math 2: 253–267
Maffray F (1992) Kernel in perfect line graph. J Comb Theor Ser B 55: 1–8
McVitie DG, Wilson LB (1970) Stable marriage assignement for unequal sets. BIT 10: 295–309
Meggido N, Papadimitriou CH (1989) A note on total functions, existence theorems, and computational complexity. Technical report, IBM
Monma CL, Wei VK (1986) Intersection graphs of paths in a tree. J Combin Theor Ser B 41: 141–181
Papadimitriou C (1994) On the complexity of the parity argument and other inefficient proofs of existence. J Comput Syst Sci 48: 498–532
Richardson M (1953) Solutions of irreflexive relations. Ann Math 58: 573–590
Schrijver A (2003) Combinatorial optimization. Springer, Berlin
Tarjan RE (1985) Decomposition by clique separators. Discret Math 55: 221–232
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The first, third, fourth and fifth authors were students at the Ecole Polytechnique when they did this work.
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de Gevigney, O.D., Meunier, F., Popa, C. et al. Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree. 4OR-Q J Oper Res 9, 175–188 (2011). https://doi.org/10.1007/s10288-010-0150-8
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DOI: https://doi.org/10.1007/s10288-010-0150-8