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Journal of Applied and Industrial Mathematics

, Volume 8, Issue 4, pp 453–457 | Cite as

Complexity of the weighted max-cut in Euclidean space

Article

Abstract

The Max-Cut Problem is considered in an undirected graph whose vertices are points of a q-dimensional Euclidean space. The two cases are investigated, where the weights of the edges are equal to (i) the Euclidean distances between the points and (ii) the squares of these distances. It is proved that in both cases the problem is NP-hard in the strong sense. It is also shown that under the assumption P≠=NP there is no fully polynomial time approximation scheme (FPTAS).

Keywords

graph cut Euclidean space NP-hard problem 

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References

  1. 1.
    A. V. Kel’manov and A. V. Pyatkin, “On the Complexity of a Search for a Subset of “Similar” Vectors,” Dokl. Ross. Akad. Nauk 421(5), 590–592 (2008) [Dokl.Math. 78 (1), 574–575 (2008)].MathSciNetGoogle Scholar
  2. 2.
    A. V. Kel’manov and A. V. Pyatkin, “NP-Completeness of Some Problems of Choosing a Vector Subset,” Diskret. Anal. Issled. Oper. 17(5), 37–45 (2010) [J. Appl. Industr.Math. 5 (3), 352–357 (2011)].MATHGoogle Scholar
  3. 3.
    A. V. Kel’manov and A. V. Pyatkin, “On Complexity of Some Problems of Cluster Analysis of Vector Sequences,” Diskret. Anal. Issled. Oper. 20(2), 47–57 (2013) [J. Appl. Industr. Math. 7 (3), 363–369 (2013)].MathSciNetGoogle Scholar
  4. 4.
    F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, “An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design,” Oper. Res. 36, 493–513 (1988).CrossRefMATHGoogle Scholar
  5. 5.
    M. Bern and D. Eppstein, “Approximation Algorithms for Geometric Problems,” in Approximation Algorithms for NP Hard Problems (PWS Publ., Boston, 1997), pp. 296–345.Google Scholar
  6. 6.
    T. N. Bui, S. Chaudhuri, F. T. Leighton, and M. Sipser, “Graph Bisection Algorithms with Good Average Case Behavior,” Combinatorica 7(2), 171–191 (1987).CrossRefMathSciNetGoogle Scholar
  7. 7.
    C. H. Q. Ding, X. He, H. Zha, M. Gu, and H. D. Simon, “AMin-Max Cut Algorithm for Graph Partitioning and Data Clustering,” in Proceedings of IEEE International Conference on Data Mining (San Jose, November 29–December 2, 2001) (IEEE Comput. Soc., Los Alamitos, 2001), pp. 107–114.CrossRefGoogle Scholar
  8. 8.
    A. Eisenblätter A. “The Semidefinite Relaxation of the k-Partition Polytope is Strong,” in Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization (Cambridge, MA, May 27–29, 2002) (Springer, Berlin, 2002), pp. 273–290.CrossRefGoogle Scholar
  9. 9.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NPCompleteness (Freeman, San Francisco, 1979).Google Scholar
  10. 10.
    M. R. Garey, D. S. Johnson, and L. Stockmeyer, “Some Simplified NP-Complete Graph Problems,” Theor. Comput. Sci. 3(1), 237–267 (1976).CrossRefMathSciNetGoogle Scholar
  11. 11.
    R. M. Karp, “Reducibility among Combinatorial Problems,” in Complexity of Computer Computations (Plenum Press, New York, 1972), pp. 85–103.CrossRefGoogle Scholar
  12. 12.
    R. M. Karp, “The Genomics Revolution and Its Challenges for Algorithmic Research,” in Current Trends in Theoretical Computer Science: Entering the 21st Century (World Sci. Publ., River Edge, New Jersey, 2001), pp. 631–642.Google Scholar
  13. 13.
    F. Liers, M. Jünger, G. Reinelt, and G. Rinaldi, “Computing Exact Ground States of Hard Ising Spin Glass Problems by Branch-and-Cut,” in New Optimization Algorithms in Physics (Wiley-VCH, Weinheim, 2004), pp. 47–68.Google Scholar
  14. 14.
    R. Lupton, M. P. Maley, and N. E. Young, “Data Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic,” J. Algorithms 27(2), 339–356 (1998).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    S. De Sousa, Y. Haxhimusa, and W. G. Kropatsch, “Estimation of Distribution Algorithm for the Max-Cut Problem,” in Proceedings of the 9th IAPR-TC-15 International Workshop Graph-Based Representations in Pattern Recognition (Vienna, Austria, May 15–17, 2013) (Springer, Heidelberg, 2013), pp. 244–253.CrossRefGoogle Scholar
  16. 16.
    V. V. Vazirani, Approximation Algorithms (Springer, Berlin, 2001).Google Scholar
  17. 17.
    F. De la Vega and C. Kenyon, “A Randomized Approximation Scheme for Metric Max-Cut,” J. Comput. Syst. Sci. 63., 531–541 (2001).CrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • A. A. Ageev
    • 1
  • A. V. Kel’manov
    • 1
    • 2
  • A. V. Pyatkin
    • 1
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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