Abstract
We consider a weighted version of the well-known Vertex Coloring Problem (VCP) in which each vertex i of a graph G has associated a positive weight w i . Like in VCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used. While in VCP the cost of each color is equal to one, in the Weighted Vertex Coloring Problem (WVCP) the cost of each color depends on the weights of the vertices assigned to that color, and it equals the maximum of these weights. WVCP is known to be NP-hard and arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose three alternative Integer Linear Programming (ILP) formulations for WVCP: one is used to derive, dropping integrality requirement for the variables, a tight lower bound on the solution value, while a second one is used to derive a 2-phase heuristic algorithm, also embedding fast refinement procedures aimed at improving the quality of the solutions found. Computational results on a large set of instances from the literature are reported.
Similar content being viewed by others
References
Boudhar, M., Finke, G.: Scheduling on a batch machine with job compatibilities. Belg. J. Oper. Res. Stat. Comput. Sci. 40, 874–885 (2000)
Brélaz, D.: New methods to color the vertices of a graph. Commun. ACM 22, 251–256 (1979)
Caprara, A., Fischetti, M., Toth, P.: A heuristic method for the set covering problem. Oper. Res. 47, 730–743 (1999)
Carpaneto, G., Martello, S., Toth, P.: Algorithms and codes for the assignment problem. Ann. Oper. Res. 13, 193–223 (1988)
De Werra, D., Demange, M., Monnot, J., Paschos, V.T.: Time slot scheduling of compatible jobs. J. Sched. 10, 111–127 (2007)
Dongarra, J.J.: Performance of various computers using standard linear equations software, (Linpack Benchmark report). Technical Report CS-89-85, University of Tennessee, Computer Science Department (2006)
Escoffier, B., Monnot, J., Paschos, V.T.: Weighted coloring: further complexity and approximability results. Inf. Process. Lett. 94, 98–103 (2006)
Finke, G., Jost, V., Queyranne, M., Sebö, A.: Batch processing with interval graph compatibilities between tasks. Cahiers du laboratoire Leibniz 108 (2004)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. Freedman, New York (1979)
Gavranovich, H., Finke, G.: Graph partitioning and set covering for the optimal design of a production system in the metal industry. In: Proceedings of the Second Conference on Management and Control of Production and Logistics (MCPL’2000), vol. 2, pp. 603–608, Grenoble, France (2000)
Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting stock problem. Oper. Res. 9, 849–859 (1961)
Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting stock problem—Part II. Oper. Res. 11, 863–888 (1963)
Hochbaum, D.S., Landy, D.: Scheduling semiconductor burn-in operations to minimize total flowtime. Oper. Res. 45, 874–885 (1997)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Johnson, D.S., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability: 2nd DIMACS Implementation Challange, 1993. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence (1996)
Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by simulated annealing: an experimental evaluation; Part II, graph coloring and number partitioning. Oper. Res. 39, 378–406 (1991)
Lodi, A., Martello, S., Vigo, D.: Models and bounds for two-dimensional level packing problems. J. Comb. Optim. 8, 363–379 (2004)
Malaguti, E., Monaci, M., Toth, P.: A metaheuristic approach for the vertex coloring problem. INFORMS J. Comput. 20, 302–316 (2008)
Mehrotra, A., Trick, M.A.: A column generation approach for graph coloring. INFORMS J. Comput. 8, 344–354 (1996)
Monaci, M., Toth, P.: A set-covering based heuristic approach for bin-packing problems. INFORMS J. Comput. 18, 71–85 (2006)
Prais, M., Ribeiro, C.C.: Reactive GRASP: An application to a matrix decomposition problem in TDMA traffic assignment. INFORMS J. Comput. 12, 164–176 (2000)
Ribeiro, C.C., Minoux, M., Penna, M.C.: An optimal column-generation-with-ranking algorithm for very large scale set partitioning problems in traffic assignment. Eur. J. Oper. Res. 41, 232–239 (1989)
Trick, M.A.: Computational symposium: Graph coloring and its generalizations. Cornell University, Ithaca, NY, 2002. http://mat.gsia.cmu.edu/COLOR02/
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Malaguti, E., Monaci, M. & Toth, P. Models and heuristic algorithms for a weighted vertex coloring problem. J Heuristics 15, 503–526 (2009). https://doi.org/10.1007/s10732-008-9075-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-008-9075-1