Complexity of the weighted max-cut in Euclidean space
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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).
Keywordsgraph cut Euclidean space NP-hard problem
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- 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
- 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
- 9.M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NPCompleteness (Freeman, San Francisco, 1979).Google Scholar
- 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.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
- 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.V. V. Vazirani, Approximation Algorithms (Springer, Berlin, 2001).Google Scholar