Abstract
Finding “good” cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP-hard in general graphs. This result holds both for the variable cardinality case (i.e., all cycles of the graph are considered) and the fixed cardinality case (i.e., only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyze rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a complete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for bounded cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akinc, U. and K.N. Srikanth. (1992). “Optimal Routing and Process Scheduling for a Mobile Service Facility.” Networks 22, 163–183.
Arkin, E.M. and R. Hassin. (1994). “Approximation Algorithms for the Geometric Covering Salesman Problem.” Discrete Applied Mathematics 55, 197–218.
Buckley, F. and F. Harary. (1990). Distance in Graphs. Reading, MA: Addison-Wesley.
Current, J.R. and D.A. Schilling. (1989). “The Covering Salesman Problem.” Transportation Science 23, 208–213.
Current, J.R. and D.A. Schilling. (1994). “The Median Tour and Maximal Covering Tour Problems: Formulations and Heuristics.” European Journal of Operational Research 73, 114–126.
Díaz-Bánez, J.M., J.A. Mesa, and A. Schöbel. (2002). “Continuous Location of Dimensional Structures.” European Journal of Operational Research. To appear.
Drezner, Z., S. Steiner, and G.O. Wesolowsky. (1996). “On the Circle Closest to a Set of Points.” Technical Report, California State University, Department of Management Science and Information Systems.
Foulds. L.R. (1998). Graph Theory Applications. Berlin: Springer.
Foulds, L.R., J.M. Wilson, and T. Yamaguchi. (1999). “Modelling and Solving Central Cycle Problems with Integer Programming.” Technical Report 1999–07, Department of Management Systems, University of Waikato, New Zealand.
Garey, M.R. and D.S. Johnson. (1979). Computers and Intractability — A Guide to the Theory of NPCompleteness. San Francisco: Freeman.
Jacobsen, S.K. and Madsen O. (1980). “A Comparative Study of Heuristics for a Two-Level Location-Routing Problem.” European Journal of Operational Research 5, 378–387.
Labbé, M. and G. Laporte. (1986). “Maximizing User Convenience and Postal Service Efficiency in Post Box Location.” Belgian Journal of Operations Research Statistics and Computer Science 26, 21–35.
Labbé, M., G. Laporte, and I. Rodriguez-Martin. (1998). “Path, Tree and Cycle Location.” In T.C. Crainic and G. Laporte (eds.), Fleet Management and Logistics. Boston: Kluwer, pp. 187–204.
Labbé, M., G. Laporte, I. Rodriguez Martin, and J.J. Salazar. (2001a). “The Median Cycle Problem.” Technical Report 12, Université Libre de Bruxelles.
Labbé, M., G. Laporte, I. Rodriguez-Martin, and J.J. Salazar. (2001b). “The Ring Star Problem: Polyhedral Analysis and Exact Algorithm.” Technical Report, Université Libre de Bruxelles.
Le, V.-B. and D.T. Lee. (1991). “Out-of-Roundness Problem Revisited.” IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 217–223.
Mesa, J.A. and T.B. Boffey. (1996). “A Review of Extensive Facility Location in Networks.” European Journal of Operational Research 95(3), 592–603.
Rodriguez Martin, I. (2000). “Cycle Location Problems.” Ph.D. Thesis, Universidad de La Laguna, Spain.
Schöbel, A. (1999). “Locating Lines and Hyperplanes –Theory and Algorithms.” Applied Optimization Series, Vol. 25. Dordrecht: Kluwer Academic.
Späth, H. (1997a). “Least Squares Fitting of Ellipses and Hyperbolas.” Computational Statistics 12(3), 329–341.
Späth, H. (1997b). “Orthogonal Distance Fitting by Circles and Ellipses.” Computational Statistics 12(3), 343–354.
Ventura, J.A. and S. Yeralan. (1989). “The Minmax Center Estimation Problem.” European Journal of Operational Research 41, 64–72.
Witzgall, C., S.I. Gass, H.H. Harary, and D.R. Shier. (August 1997). “Linear Programming Techniques for Fitting Circles and Spheres.” In ISMP'97, Lausanne.
Yamaguchi, T., L.R. Foulds, and J. Lamb. (1998). “Central Cycles in Graphs.” MIT Memoirs 48, 93–99.
Yeralan, S. and J.A. Ventura. (1988). “Computerized Roundness Inspection.” International Journal of Production Research 26, 1921–1935.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Foulds, L.R., Hamacher, H.W., Schöbel, A. et al. On Center Cycles in Grid Graphs. Annals of Operations Research 122, 163–175 (2003). https://doi.org/10.1023/A:1026198523981
Issue Date:
DOI: https://doi.org/10.1023/A:1026198523981