# The Optimal Cost Chromatic Partition problem for trees and interval graphs

## 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

- 1.E.M. Arkin, and E.L. Silverberg. Scheduling jobs with fixed start and finish times.
*Discrete Applied Mathematics*, 18 (1987), 1–8.Google Scholar - 2.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).Google Scholar - 3.V. Chvatal. On certain polytopes associated with graphs.
*J. Comb. Theory*, B-18 (1975) 138–154.Google Scholar - 4.V.R. Dondeti, and H. Emmons. Job scheduling with processors of two types.
*Operations Research*, 40 (1992) S76–S85.Google Scholar - 5.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.Google Scholar - 6.M. Fischetti, S. Martello, and P. Toth. The Fixed Job Schedule Problem with spread time constraints.
*Operations Research*, 6 (1987) 849–858.Google Scholar - 7.M. Fischetti, S. Martello, and P. Toth. The Fixed Job Schedule Problem with working time constraints.
*Operations Research*, 3 (1989) 395–403.Google Scholar - 8.M. Fischetti, S. Martello and P. Toth. Approximation algorithms for Fixed Job Schedule Problems.
*Operations Research*, 40 (1992) S96–S108.Google Scholar - 9.M.R. Garey, and D.S. Johnson.
*Computers and intractability: A guide to the theory of NP-Completeness*. Freeman, San Fransisco, 1979.Google Scholar - 10.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.Google Scholar - 11.C. Grinstead. The perfect graph conjecture for toroidal graphs.
*Annals of Discrete Mathematics*, 21 (1984) 97–101.Google Scholar - 12.M. Grötschel, L. Lovasz, and A. Schrijver.
*Geometric algorithms and combinatorial optimization*, (1988). Springer Verlag.Google Scholar - 13.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.Google Scholar - 14.A.W.J. Kolen, and L.G. Kroon. Licence Class Design: complexity and algorithms.
*European Journal of Operational Research*, 63 (1992), 432–444.Google Scholar - 15.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.Google Scholar - 16.A.W.J. Kolen, and L.G. Kroon. Analysis of shift class design problems.
*European Journal of Operational Research*, 79 (1994). 417–430.Google Scholar - 17.E. Kubicka, and A. Schwenk. Introduction to chromatic sums.
*Proc. ACM Computer Science Conference*, (1989).Google Scholar - 18.M.W. Padberg. Perfect Zero-One matrices.
*Mathematical Programming*, 6 (1974) 180–196.Google Scholar - 19.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.Google Scholar - 20.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.Google Scholar - 21.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.Google Scholar
- 22.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.Google Scholar - 23.A. Tucker. The Strong Perfect Graph Conjecture for planar graphs.
*Canad. J. Math*, 25 (1973) 103–114.Google Scholar - 24.A. Tucker. The validity of the perfect graph conjecture for
*K*_{4}-free graphs.*Annals of Discrete Mathematics*, 21 (1984) 149–157.Google Scholar - 25.M. Yanakakis, and F. Gavril. The maximum
*k*-colorable subgraph problem for chordal graphs.*Information Processing Letters*, 24 (1987), 133–137.Google Scholar