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Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs

  • Tomomi Matsui
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1763)

Abstract

Unit disk graphs are the intersection graphs of equal sized circles in the plane.

In this paper, we consider the maximum independent set problems on unit disk graphs. When the given unit disk graph is defined on a slab whose width is k, we propose an algorithm for finding a maximum independent set in \(\mathrm{O}(n^{4\lceil 2k/ \sqrt{3}\rceil})\) time where n denotes the number of vertices. We also propose a (1 – 1/r)-approximation algorithm for the maximum independent set problems on a (general) unit disk graph whose time complexity is bounded by \(\mathrm{O}(rn^{4\lceil 2(r-1)/ \sqrt{3}\rceil})\).

We also propose an algorithm for fractional coloring problems on unit disk graphs. The fractional coloring problem is a continuous version of the ordinary (vertex) coloring problem. Our approach for the independent set problem implies a strongly polynomial time algorithm for the fractional coloring problem on unit disk graphs defined on a fixed width slab. We also propose a strongly polynomial time 2-approximation algorithm for fractional coloring problem on a (general) unit disk graph.

Keywords

Approximation Algorithm Polynomial Time Polynomial Time Algorithm Intersection Graph Coloring Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and intractability of approximation problems. In: Proc. 33rd IEEE Symposium on Foundations of Computer Science, pp. 13–22 (1992)Google Scholar
  2. 2.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86, 165–177 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)zbMATHGoogle Scholar
  6. 6.
    Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.S.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34, 250–256 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. on Computing 6, 505–517 (1977)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Tomomi Matsui
    • 1
  1. 1.Graduate School of EngineeringUniversity of TokyoBunkyo-ku, TokyoJapan

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