Abstract
In this paper we study the Optimal Cost Chromatic Partition (OCCP) problem for trees and interval graphs. The OCCP problem is the problem of coloring the nodes of a graph in such a way that adjacent nodes obtain different colors and that the total coloring costs are minimum.
In this paper we first give a linear time algorithm for the OCCP problem for trees. The OCCP problem for interval graphs is equivalent to the Fixed Interval Scheduling Problem with machine-dependent processing costs. We show that the OCCP problem for interval graphs can be solved in polynomial time if there are only two different values for the coloring costs. However, if there are at least four different values for the coloring costs, then the OCCP problem for interval graphs is shown to be NP-hard. We also give a formulation of the latter problem as an integer linear program, and prove that the corresponding coefficient matrix is perfect if and only if the associated intersection graph does not contain an odd hole of size 7 or more as a node-induced subgraph. Thereby we prove that the Strong Perfect Graph Conjecture holds for graphs of the form K×G, where K is a clique and G is an interval graph.
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References
E.M. Arkin, and E.L. Silverberg. Scheduling jobs with fixed start and finish times. Discrete Applied Mathematics, 18 (1987), 1–8.
C. Berge. Färbung von Graphen deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss. Z. Martin-Luther Univ. Halle-Wittenberg. Math.-Natur. Reihe, 114 (1961).
V. Chvatal. On certain polytopes associated with graphs. J. Comb. Theory, B-18 (1975) 138–154.
V.R. Dondeti, and H. Emmons. Job scheduling with processors of two types. Operations Research, 40 (1992) S76–S85.
V.R. Dondeti, and H. Emmons. Algorithms for preemptive scheduling of different classes of processors to do jobs with fixed times. European Journal of Operational Research, 70 (1993) 316–326.
M. Fischetti, S. Martello, and P. Toth. The Fixed Job Schedule Problem with spread time constraints. Operations Research, 6 (1987) 849–858.
M. Fischetti, S. Martello, and P. Toth. The Fixed Job Schedule Problem with working time constraints. Operations Research, 3 (1989) 395–403.
M. Fischetti, S. Martello and P. Toth. Approximation algorithms for Fixed Job Schedule Problems. Operations Research, 40 (1992) S96–S108.
M.R. Garey, and D.S. Johnson. Computers and intractability: A guide to the theory of NP-Completeness. Freeman, San Fransisco, 1979.
M.R. Garey, D.S. Johnson, G.L. Miller and C.H. Papadimitriou. The complexity of coloring circular arcs and chords. SIAM Journal of Alg. Disc. Meth, 1 (1980) 216–227.
C. Grinstead. The perfect graph conjecture for toroidal graphs. Annals of Discrete Mathematics, 21 (1984) 97–101.
M. Grötschel, L. Lovasz, and A. Schrijver. Geometric algorithms and combinatorial optimization, (1988). Springer Verlag.
A.W.J. Kolen, and L.G. Kroon. On the computational complexity of (maximum) class scheduling. European Journal of Operational Research, 54 (1991), 23–38.
A.W.J. Kolen, and L.G. Kroon. Licence Class Design: complexity and algorithms. European Journal of Operational Research, 63 (1992), 432–444.
A.W.J. Kolen, and L.G. Kroon. On the computational complexity of (maximum) shift class scheduling. European Journal of Operational Research, 64 (1993), 138–151.
A.W.J. Kolen, and L.G. Kroon. Analysis of shift class design problems. European Journal of Operational Research, 79 (1994). 417–430.
E. Kubicka, and A. Schwenk. Introduction to chromatic sums. Proc. ACM Computer Science Conference, (1989).
M.W. Padberg. Perfect Zero-One matrices. Mathematical Programming, 6 (1974) 180–196.
K.R. Parthasarathy, and G. Ravindra. The Strong Perfect Graph Conjecture is true for K 1,3-free graphs. J. Comb. Theory, B-21 (1976) 212–223.
A. Sen, H. Deng, and S. Guha. On a graph partition problem with an application to VLSI layout. Information Processing Letters, 43 (1991) 87–94.
A. Sen, H. Deng, and A. Roy. On a graph partition problem with application to multiprocessor scheduling. TR-92-020. Department of Computer Science and Engineering, Arizona State University.
K.J. Supowit. Finding a maximum planar subset of a set of nets in a channel. IEEE Trans. on Computer Aided Design, CAD 6, 1 (1987) 93–94.
A. Tucker. The Strong Perfect Graph Conjecture for planar graphs. Canad. J. Math, 25 (1973) 103–114.
A. Tucker. The validity of the perfect graph conjecture for K 4-free graphs. Annals of Discrete Mathematics, 21 (1984) 149–157.
M. Yanakakis, and F. Gavril. The maximum k-colorable subgraph problem for chordal graphs. Information Processing Letters, 24 (1987), 133–137.
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© 1997 Springer-Verlag Berlin Heidelberg
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Kroon, L.G., Sen, A., Deng, H., Roy, A. (1997). The Optimal Cost Chromatic Partition problem for trees and interval graphs. In: d'Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds) Graph-Theoretic Concepts in Computer Science. WG 1996. Lecture Notes in Computer Science, vol 1197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62559-3_23
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DOI: https://doi.org/10.1007/3-540-62559-3_23
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