k Best Cuts for Circular-Arc graphs

  • K. -H. Tsai
  • D. T. Lee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 834)


Given a set of n weighted circular-arcs on a unit circle, and an integer k, the k Best Cuts for Circular-Arcs problem, abbreviated as k-BCCA problem, is to find a placement of k points, called cuts, 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, the k Best Cuts for Intervals (k-BCI) problem in O(kn+n log n) time. We then define the k Restricted Best Cuts for Intervals (k-RBCI) problem, and solve it in the same complexity of k-BCI algorithm. These two algorithms are then used as subroutines to solve the k-BCCA problem in O(kn+n log n) or O(I(k)+n log n) time, where I(k) is the time complexity of k-BCI problem. We also show that if k}>1, the k Maximum Cliques Cover problem for circular-arc graphs can be solved in O(I(k)+n log n) time.


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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) 195–208CrossRefGoogle Scholar
  2. 2.
    A. Aggarwal, B. Schieber, and T. Tokuyama: Finding a Minimum-Weight K-Link Path in Graphs with Monge Property and Applications. Proc. 9th Symp. on Comput. Geometry (1993) 189–197Google Scholar
  3. 3.
    A. V. Aho, J. E. Hopcroft and J. D. Ullman: The Design and Analysis of Computer Algorithms. Addison-Wesley, (1974)Google Scholar
  4. 4.
    M. Ben-Or: Lower bounds for algebraic computation trees. Proc. 15th Annual Symp. on Theory of Computing (1983) 80–86Google Scholar
  5. 5.
    Z. Drezner: Competitive location strategies for two facilities. Regional Science and Urban Economics 12 (1982) North-Holland, 485–493Google Scholar
  6. 6.
    S. L. Hakimi: On locating new facilities in a competitive environment. ISOLDE II June (1981) Skodsborg, Denmark.Google Scholar
  7. 7.
    W.-L. Hsu and K. H. Tsai: Linear time algorithms on circular-arc graphs. Proc. 26th Allerton Conference on Communication, Control and Computing (1988) 842–851Google Scholar
  8. 8.
    H. N. Gabow and R. E. Tarjan: A linear-time algorithm for a special case of disjoint set union. Proc. 15th ACM Symposium on Theory of Computing (Boston, Mass, Apr. 25–27) ACM New York (1983) 246–251Google Scholar
  9. 9.
    D.T. Lee: Maximum clique problem of rectangle graphs. Advances in Computing Research, F. P. Preparata, ed., JAI Press (1983), 91–107Google Scholar
  10. 10.
    D. T. Lee, M. Sarrafzadeh and Y.F. Wu: Minimum Cut for Circular-arc Graphs. SIAM J. Computing. 19(6) Dec. (1990) 1041–1050CrossRefGoogle Scholar
  11. 11.
    D. T. Lee and Y. F. Wu: Geometric complexity of some location problems. Algorithmica, 1 (1985) 193–211Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

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

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