Advertisement

Annals of Operations Research

, Volume 275, Issue 2, pp 281–295 | Cite as

On interval and circular-arc covering problems

  • Reuven Cohen
  • Mira GonenEmail author
Original Research
  • 40 Downloads

Abstract

In this paper we study several related problems of finding optimal interval and circular-arc covering. We present solutions to the maximum k-interval (k-circular-arc) coverage problems, in which we want to cover maximum weight by selecting k intervals (circular-arcs) out of a given set of intervals (circular-arcs), respectively, the weighted interval covering problem, in which we want to cover maximum weight by placing k intervals with a given length, and the k-centers problem. The general sets version of the discussed problems, namely the general measure k-centers problem and the maximum covering problem for sets are known to be NP-hard. However, for the one dimensional restrictions studied here, and even for circular-arc graphs, we present efficient, polynomial time, algorithms that solve these problems. Our results for the maximum k-interval and k-circular-arc covering problems hold for any right continuous positive measure on \(\mathbb {R}\).

Keywords

Covering Optimization Dynamic programming 

Notes

Acknowledgements

Reuven Cohen thanks the BSF for support. Science and Technology of Israel.

References

  1. Agarwal, P. K., & Procopiuc, C. M. (2002). Exact and approximation algorithms for clustering. Algorithmica, 33, 201–226.CrossRefGoogle Scholar
  2. Ageev, A. A. & Sviridenko, M. I. (1999). Approximation algorithms for maximum coverage and max cut with given sizes of parts. In Proceedings of the IPCO (pp. 17–30).Google Scholar
  3. Alon, N., Moshkovitz, D., & Safra, S. (April 2006). Algorithmic construction of sets for k-restrictions. In ACM Transactions on Algorithms (TALG) (pp. 153–177).Google Scholar
  4. Bar-Ilan, J., & Peleg, D. (1991). Approximation algorithms for selecting network centers. In Proceedings of the 2nd workshop on algorithms and data structures, lecture notes in computer science (pp. 343–354).Google Scholar
  5. Brass, P., Knauer, C., Na, H. S., Shin, C. S., & Vigneron, A. (2009). Computing \(k\)-centers on a line. arXiv:0902.3282v1.
  6. Brönnimann, H., & Goodrich, M. T. (1995). Almost optimal set covers in finite VC dimension. Discrete & Computational Geometry, 14, 263–279.CrossRefGoogle Scholar
  7. Caprara, A., & Toth, P. (2000). Algorithms for the set covering problem. Annals of Operations Research, 98, 353–371.CrossRefGoogle Scholar
  8. Carmi, P., Katz, M. J., & Lev-Tov, N. (2007). Covering points by unit disks of fixed location. In Proceedings of the 18th International Symposium on Algorithms and Computation (ISAAC) (pp. 644–655).Google Scholar
  9. Chakrabarty, D., Grant, E., & Köenemann, J. (2010). On column-restricted and priority covering integer programs. In Integer programming and combinatorial optimization (pp. 355–368).Google Scholar
  10. Chan, T. (1999). Geometric applications of a randomized optimization technique. Discrete & Computational Geometry, 22, 547–567.CrossRefGoogle Scholar
  11. Chan, T. M., & Grant, E. (2014). Exact algorithms and APX-hardness results for geometric packing and covering problems. Computational Geometry, 47, 112–124.CrossRefGoogle Scholar
  12. Chen, D. Z., & Wang, H. (2011). Efficient algorithms for the weighted \(k\)-center problem on a real line. In Proceedings of the 22nd International Symposium on Algorithms and Computation (ISAAC) (pp. 584–593).Google Scholar
  13. Cohen, R., Gonen, M., Levin, A., & Onn, S. (2017). On nonlinear multi-covering problems. Journal of Combinatorial Optimization, 33(2), 645–659.CrossRefGoogle Scholar
  14. Cornuejols, G., Nemhauser, G. L., & Wolsey, L. A. (1980). Worst-case and probabilistic analysis of algorithms for a location problem. Operations Research, 28, 847–858.CrossRefGoogle Scholar
  15. de Werraa, D., Eisenbeisb, C., Lelaitc, S., & Stöhr, E. (2002). Circular-arc graph coloring: On chords and circuits in the meeting graph. European Journal of Operational Research, 136, 483–500.CrossRefGoogle Scholar
  16. Eppstein, D. (1997). Fast construction of planar two-centers. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 131–138).Google Scholar
  17. Erlebach, T., & van Leeuwen, E. J. (2010). PTAS for weighted set cover on unit squares. In Proceedings of the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) and of the 14th International Workshop on Randomization and Computation (RANDOM) (pp. 166–177).Google Scholar
  18. Even, G., Rawitz, D., & Shahar, S. (2005). Hitting sets when the VC-dimension is small. Information Processing Letters, 95, 358–362.CrossRefGoogle Scholar
  19. Feige, Uriel. (1998). A threshold of ln n for approximating set cover. Journal of the ACM, 45(4), 634–652.CrossRefGoogle Scholar
  20. Fowler, R., Paterson, M., & Tanimoto, S. (1981). Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12, 133–137.CrossRefGoogle Scholar
  21. Garey, M. R., & Johnson, D. S. (1978). Computers and intractability: A guide to the theory of NP-completeness. New York: Freeman.Google Scholar
  22. Gonzalez, T. F. (1985). Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38, 293–306.CrossRefGoogle Scholar
  23. Gonzalez, T. F. (1991). Covering a set of points in multidimensional space. Information Processing Letters, 40, 181–188.CrossRefGoogle Scholar
  24. HallRakesh, N. G., & Vohra, V. (1993). Pareto optimality and a class of set covering heuristics. Annals of Operations Research, 43, 279–284.CrossRefGoogle Scholar
  25. Hochbaum, Dorit S., & Levin, Asaf. (2006). Optimizing over consecutive 1’s and circular 1’s constraints. SIAM Journal on Optimization, 17(2), 311–330.CrossRefGoogle Scholar
  26. Hochbaum, D. S., & Maass, W. (1987). Fast approximation algorithms for a nonconvex covering problem. Journal of Algorithms, 8, 305–323.CrossRefGoogle Scholar
  27. Hochbaum, D. S., & Shmoys, D. B. (1986). A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM, 33, 533–550.CrossRefGoogle Scholar
  28. Hsu, W. L., & Nemhauser, G. L. (1979). Easy and hard bottleneck location problems. Discrete Applied Mathematics, 1, 209–216.CrossRefGoogle Scholar
  29. Hwang, R. Z., Lee, R. C. T., & Chang, R. C. (1993). The slab dividing approach to solve the Euclidean \(p\)-center problem. Algorithmica, 9, 1–22.CrossRefGoogle Scholar
  30. Lovász, L. (1975). On the ratio of optimal integral and fractional covers. SIAM Journal on Discrete Mathematics, 13, 383–390.CrossRefGoogle Scholar
  31. Masuyama, S., Ibaraki, T., & Hasegawa, T. (1981). The computational complexity of the m-centers problem on the plane. Transactions IECE of Japan, E64, 57–64.Google Scholar
  32. Megiddo, N. (1990). On the complexity of some geometric problems in unbounded dimension. Journal of Symbolic Computation, 10, 327–334.CrossRefGoogle Scholar
  33. Megiddo, N., & Supowit, K. (1984). On the complexity of some common geometric location problems. SIAM Journal on Computing, 13, 182–196.CrossRefGoogle Scholar
  34. Mustafa, N. H., & Ray, S. (2010). Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44, 883–895.CrossRefGoogle Scholar
  35. Papadimitriou, C. H. (1994). Computational Complexity. Boston: Addison-Wesley.Google Scholar
  36. Plesník, J. (1980). On the computational complexity of centers locating in a graph. Aplikace Matematiky, 25, 445–452.Google Scholar
  37. Plesnk, J. (1980). A heuristic for the p-center problem in graphs. Discrete Applied Mathematics, 17, 263–268.CrossRefGoogle Scholar
  38. Raz, R., & Safra, S. (1997). A sub-constant error-probability PCP characterization of NP. In Proceedings of the 29th Symposium on the Theory of Computing (STOC) (pp. 475–484).Google Scholar
  39. Renata, K., & KwateraBruno, S. (1993). Clustering heuristics for set covering. Annals of Operations Research, 43, 295–308.CrossRefGoogle Scholar
  40. Revelle, C., & Hogan, K. (1989). The maximum reliability location problem and \(\alpha \)-reliable p-center problem: Derivatives of the probabilistic location set covering problem. Annals of Operations Research, 18, 155–173.CrossRefGoogle Scholar
  41. Slavik, P. (May 1995). Improved approximations of packing and covering problems. In Proceedings of the 27th Symposium on the Theory of Computing (STOC), pp. 268–276, Baltimore, MD, USA.Google Scholar
  42. Srinivasan, A. (1999). Improved approximations guarantees for packing and covering integer programs. SIAM Journal on Computing, 29(2), 648–670.CrossRefGoogle Scholar
  43. Supowit, K.J. (1981). Topics in computational geometry. In Technical Report UIUCDCS-R-81-1062 Urbana, IL: Department of Computer Science, University of Illinois.Google Scholar
  44. Vercellis, C. (1984). A probabilistic analysis of the set covering problem. Annals of Operations Research, 1, 255–271.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.Department of Computer ScienceAriel UniversityArielIsrael

Personalised recommendations