L. G. Kroon, A. Sen, H. Deng and A. Roy, The Optimal Cost Chromatic Partition Problem for Trees and Interval Graphs, In Proc. of Workshop on Graph-Theoretical Concepts in Computer Science (WG'96), Lecture Notes in Computer Science, Springer, 1197, pp. 279--292, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....coloring problems are NPhard since the corresponding ordinary coloring problems are NP hard, respectively. However, it has been known that the cost vertex and edge coloring problems can be solved in polynomial time for trees: the cost vertex coloring problem can be solved in time O(n) for trees [KSDR97], and the cost edge coloring problem can be solved ) for trees [ZN01] where Delta is the maximum degree. In this thesis, we prove in Chapter 5 that the cost total coloring problem can be solved in time O(n Delta for trees. 1.4 Summary In this section, we briefly summerize our main ....

....arboricity a Delta 8a 1 Delta 1 O(an thickness Delta 24 1 Delta 1 O( n genus fl 1 Delta 4 Xi (5 p 48fl 1) 2 Pi 3 Delta 1 O( p fln Table 1.3: Results on total colorings of various classes of graphs. colorings graphs time references cost vertex trees O(n) [KSDR97] cost vertex interval graphs NP hard [KSDR97] cost edge trees O(n Delta ) ZN01] cost total trees O(n Delta Table 1.4: Results on the cost colorings. Preliminaries In this chapter we formally present basic and standard terminologies and notations on graph theory which will be used in ....

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L. G. Kroon, A. Sen, H. Deng and A. Roy, The Optimal Cost Chromatic Partition Problem for Trees and Interval Graphs, In Proc. of Workshop on Graph-Theoretical Concepts in Computer Science (WG'96), Lecture Notes in Computer Science, Springer, 1197, pp. 279--292, 1997.

....to execute job v on machine c. The problem is to nd a partition of the graph G into independent sets U 1 ; Um such that P m c=1 P v2Uc k v;c is minimum. A subproblem with only machine dependent costs k c = k v;c for each v 2 V and m = n, called the OCCP problem, has been studied in [16, 10, 9]. In this paper, we consider the restricted case with m = n and costs k v;c = k v;3 for c 3 and v 2 V . We prove that the polyhedron corresponding to the ILP contains only integral 0=1 extrema if and only if the graph G in the instance is a parity graph. The OCCP problem restricted to circle ....

....to the ILP contains only integral 0=1 extrema if and only if the graph G in the instance is a parity graph. The OCCP problem restricted to circle and permutation graphs, introduced by Supowit [17] corresponds to a VLSI layout problem (see also [16] Another application is given by Kroon et al. [10]. The OCCP problem for interval graphs is equivalent to the Fixed Interval Scheduling Problem (FISP) with machine dependent processing costs. In this scheduling problem each job j 2 J must be executed during a given time interval (s j ; f j ) We assume that a sucient number of machines is ....

[Article contains additional citation context not shown here]

L.G. Kroon, A. Sen, H. Deng and A. Roy, The optimal cost chromatic partition problem for trees and interval graphs, Graph Theoretical Concepts in Computer Science WG 96, Como, LNCS (1996).

....on di erent layers in such a way that routing segments do not cross on the same layer and that the total costs are minimum. Therefore, this layout problem is equivalent to the OCCP problem (restricted e.g. to circle graphs or to permutation graphs) Another application is given by Kroon et al. [24]. The OCCP problem for interval graphs is equivalent to the Fixed Interval Scheduling Problem (FISP) with machine dependent processing costs. In this scheduling problem each job j 2 J must be executed during a given time interval (s j ; f j ) We assume that a sucient number of machines is ....

....corresponding to the constraints of this ILP contains only integral (0 1) vertices, if the cartesian product G K jV j is a perfect graph. Sen et al. 27] proved that G K jV j is perfect if G is a tree and that there exists a perfect graph G such that G K jV j is not perfect. Kroon et al. [24] studied the OCCP problem for interval graphs and trees. They showed that the problem restricted to trees can be solved in linear time and that the problem restricted to interval graphs is NP complete even if there are only four di erent values for the coloring costs. For interval graphs G, they ....

[Article contains additional citation context not shown here]

L.G. Kroon, A. Sen, H. Deng and A. Roy, The optimal cost chromatic partition problem for trees and interval graphs, to appear in: Workshop on Graph Theoretical Concepts in Computer Science (1996).

....on the same layer and that the total costs are minimum. Therefore, this layout problem is equivalent to the OCCP problem (restricted e.g. to circle graphs or to permutation graphs) 1991 Mathematics Subject Classi cation. 05 C 15. 1 2 KLAUS JANSEN Another application is given by Kroon et al. [19]. The OCCP problem for interval graphs is equivalent to the Fixed Interval Scheduling Problem (FISP) with machine dependent processing costs. In this scheduling problem each job j 2 J must be executed during a given time interval (s j ; f j ) We assume that a sucient number of machines is ....

....corresponding to the constraints of this ILP contains only integral (0 1) vertices if the cartesian product G K jV j is a perfect graph. Sen et al. 21] proved that G K jV j is perfect if G is a tree and that there exists a perfect graph G such that G K jV j is not perfect. Kroon et al. [19] studied the OCCP problem for interval graphs and trees. They showed that the problem restricted to trees can be solved in linear time and that the problem restricted to interval graphs is NP complete even if there are only four di erent values for the coloring costs. If there are only two ....

[Article contains additional citation context not shown here]

L.G. Kroon, A. Sen, H. Deng and A. Roy, The optimal cost chromatic partition problem for trees and interval graphs, to appear in: Workshop on Graph Theoretical Concepts in Computer Science (1996).

....to execute job v on machine c. The problem is to find a partition of the graph G into independent sets U 1 ; Um such that P m c=1 P v2Uc k v;c is minimum. A subproblem with only machine dependent costs k c = k v;c for each v 2 V and m = n, called the OCCP problem, has been studied in [16, 10, 9]. In this paper, we consider the restricted case with m = n and costs k v;c = k v;3 for c 3 and v 2 V . We prove that the polyhedron corresponding to the ILP contains only integral 0=1 extrema if and only if the graph G in the instance is a parity graph. The OCCP problem restricted to circle and ....

....to the ILP contains only integral 0=1 extrema if and only if the graph G in the instance is a parity graph. The OCCP problem restricted to circle and permutation graphs, introduced by Supowit [17] corresponds to a VLSI layout problem (see also [16] Another application is given by Kroon et al. [10]. The OCCP problem for interval graphs is equivalent to the Fixed Interval Scheduling Problem (FISP) with machine dependent processing costs. In this scheduling problem each job j 2 J must be executed during a given time interval (s j ; f j ) We assume that a sufficient number of machines is ....

[Article contains additional citation context not shown here]

L.G. Kroon, A. Sen, H. Deng and A. Roy, The optimal cost chromatic partition problem for trees and interval graphs, Graph Theoretical Concepts in Computer Science WG 96, Como, LNCS (1996).

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