, Volume 18, Issue 2, pp 198–216 | Cite as

k best cuts for circular-arc graphs

  • K. H. Tsai
  • D. T. Lee


Given a set ofn nonnegativeweighted circular arcs on a unit circle, and an integerk, thek Best Cust for Circular-Arcs problem, abbreviated as thek-BCCA problem, is to find a placement ofk points, calledcuts, on the circle such that the total weight of the arcs that contain at least one cut is maximized.

We first solve a simpler version, thek Best Cuts for Intervals (k-BCI) problem, inO(kn+n logn) time andO(kn) space using dynamic programming. The algorithm is then extended to solve a variation, called thek-restricted BCI problem, and the space complexity of thek-BCI problem can be improved toO(n). Based on these results, we then show that thek-BCCA problem can be solved inO(I(k,n)+nlogn) time, whereI(k, n) is the time complexity of thek-BCI problem. As a by-product, thek Maximum Cliques Cover problem (k>1) for the circular-arc graphs can be solved inO(I(k,n)+nlogn) time.

Key Words

Circular-arc graph Interval graph Facility location Competitive location Maximum clique cover 


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  1. [1]
    A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilbur, Geometric applications of a matrix-searching algorithm,Algorithmica,2 (1987), 195–208.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Aggarwal, B. Schieber, and T. Tokuyama, Finding a minimum-weightK-link path in graphs with Monge property and applications,Proc. 9th Symp. on Computational Geometry, 1993, pp. 189–197.Google Scholar
  3. [3]
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA, 1974.zbMATHGoogle Scholar
  4. [4]
    M. Ben-Or, Lower bounds for algebraic computation trees,Proc. 15th Annual Symp. on Theory of Computing, 1983, pp. 80–86.Google Scholar
  5. [5]
    Z. Drezner, Competitive location strategies for two facilities,Regional Sci. Urban Econ.,12 (1982), 485–493.CrossRefGoogle Scholar
  6. [6]
    H. N. Gabow and R. E. Tarjan, A linear-time algorithm for a special case of disjoint set union,Proc. 15th ACM Symp. on Theory of Computing, Boston, MA, Apr. 25–27, 1983, pp. 246–251.Google Scholar
  7. [7]
    S. L. Hakimi, On locating new facilities in a competitive environment,ISOLDE II, June 1981, Skodsborg, Denmark.Google Scholar
  8. [8]
    W.-L. Hsu and K. H. Tsai, Linear time algorithms on circular-arc graphs,Proc. 26th Allerton Conf. on Communication, Control and Computing, 1988, pp. 842–851.Google Scholar
  9. [9]
    D. T. Lee, Maximum clique problem of rectangle graphs, InAdvances in Computing Research, F. P. Preparata, ed., JAI Press, Greenwich, CT, 1983, pp. 91–107.Google Scholar
  10. [10]
    D. T. Lee, M. Sarrafzadeh and Y. F. Wu, Minimum cut for circular-arc graphs,SIAM J. Comput.,19(6) (1990), 1041–1050.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    D. T. Lee and Y. F. Wu, Geometric complexity of some location problems,Algorithmica,1 (1985), 193–211.MathSciNetCrossRefGoogle Scholar
  12. [12]
    A. S. Rao and C. P. Rangan, Optimal parallel algorithms on circular arc graphs,Inform. Process. Lett.,33 (1989), 147–156.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Y. F. Wu, Geometric Location and Distance Problems, Ph.D. Dissertation, Dept. of EE/CS, Northwestern University, 1985.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • K. H. Tsai
    • 1
  • D. T. Lee
    • 2
  1. 1.Institute of Information ScienceAcademia SinicaTaipeiTaiwan
  2. 2.Department of Electrical and Computer EngineeringNorthwestern UniversityEvanstonUSA

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