Abstract
Consider a set of n fixed length intervals and a set of n (larger) windows, in one-to-one correspondence with the intervals, and assume that each interval can be placed in any position within its window. If the position of each interval has been fixed, the intersection graph of such set of intervals is an interval graph. By varying the position of each interval in all possible ways, we get a family of interval graphs. In the paper we define some optimization problems related to the clique, stability, chromatic, clique cover numbers and cardinality of the minimum dominating set of the interval graphs in the family, mainly focussing on complexity aspects, bounds and solution algorithms. Some problems are proved to be NP-hard, others are solved in polynomial time on some particular classes of instances. Many practical applications can be reduced to these kind of problems, suggesting the use of Shiftable Intervals as a new interesting modeling framework.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonfiglio, P., F. Malucelli, and S. Nicoloso. (1997). “The Dominating Set Problem on Shiftable Interval Graphs.” Technical Report no. 461, IASI-CNR Rome.
Booth, K.S. and J.H. Johnson. (1982). “Dominating Sets in Chordal Graphs.” SIAM J. Comput., 11, 191–199.
Farber, M. (1984). “Domination, Independent Domination, and Duality in Strongly Chordal Graphs.” Discr. Appl. Math. 7, 115–130.
Garey, M.R. and D.S. Johnson. (1975). “Complexity Results for Multiprocessor Scheduling Under Resource Constraints.” SIAM Journal on Computing, 4(4), 397–411.
Garey, M.R. and D.S. Johnson. (1977). “Two-Processor Scheduling with Start-Times and Deadlines.” SIAM Journal on Computing, 6, 416–426.
Garey M.R. and D.S. Johnson. (1979). Computers and Intractability A Guide to the Theory of NP-Completeness, W. H Freeman and Company, New York.
Gilmore, P.C. and A.J. Hoffman. (1964). “A Characterization of Comparability and of Interval Graphs.” Canad. J. Math., 16, 539–548.
Gupta U.I., D.T. Lee, and J.Y.-T. Leung. (1982). “Efficient Algorithms for Interval Graphs and Circular Arc Graphs.” Networks, 12, 459–467.
Golumbic M.C. (1980). Algorithmic Graph Theory and Perfect Graphs, Academic Press Inc. San Diego.
Golumbic M.C. and R. Shamir. (1993). “Complexity and Algorithms for Reasoning About Time: A Graph-Theoretic Approach.” Journal of ACM, 40(5), 1108–1133.
Kise H., T. Ibaraki, and H. Mine. (1978). “A Solvable Case of the One Machine Scheduling Problem with Ready and Due Dates.” Operations Research, 26, 121–126.
Lenstra J.K., A.H.G. Rinnooy-Kan, and P. Brucker. (1977). “Complexity of Machine Scheduling Problems.” Annals of Discrete Mathematics, 1, 343–362.
Preparata F.P. and M.I. Shamos. (1985). Computational Geometry: An Introduction, Springer-Verlag, New York.
Ramalingan, G. and C.P. Rangan. (1988). “A Unified Approach to Domination Problems on Interval Graphs.” Inf. Proc. Lett., 27, 271–274.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Malucelli, F., Nicoloso, S. Shiftable intervals. Ann Oper Res 150, 137–157 (2007). https://doi.org/10.1007/s10479-006-0161-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-006-0161-1