# Maximum Dispersion and Geometric Maximum Weight Cliques

## Abstract

We consider geometric instances of the problem of finding a set of *k* vertices in a complete graph with nonnegative edge weights. In particular, we present algorithmic results for the case where vertices are represented by points in *d*-dimensional space, and edge weights correspond to rectilinear distances. This problem can be considered as a facility location problem, where the objective is to “disperse” a number of facilities, i.e., select a given number of locations from a discrete set of candidates, such that the average distance between selected locations is maximized. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where *k* is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where *k* is part of the input, we present a polynomial-time approximation scheme.

## Keywords

Edge Weight Facility Location Problem Polynomial Time Approximation Scheme Dense Subgraph Maximum Dispersion## Preview

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## References

- 1.S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for NP-hard problems. In
*Proceedings of the 27th Annual ACM Symposium on Theory of Computing*, pages 284–293, 1995.Google Scholar - 2.Y. Asahiro, R. Hassin, and K. Iwama. Complexity of finding dense subgraphs. Manuscript, 2000.Google Scholar
- 3.Y. Asahiro, K. Iwama, H. Tamaki, and T. Tokuyama. Greedily finding a dense graph. In
*Proceedings of the 5th Scandinavian Workshop on Algorithm Theory (SWAT)*, volume 1097 of*Lecture Notes in Computer Science*, pages 136–148. Springer-Verlag, 1996.Google Scholar - 4.B. Chandra and M. M. Halldórsson. Facility dispersion and remote subgraphs. In
*Proceedings of the 5th Scandinavian Workshop on Algorithm Theory (SWAT)*, volume 1097 of Lecture Notes in Computer Science, pages 53–65. Springer-Verlag, 1996.Google Scholar - 5.E. Erkut. The discrete p-dispersion problem.
*European Journal of Operations Research*, 46:48–60, 1990.MATHCrossRefMathSciNetGoogle Scholar - 6.U. Feige, G. Kortsarz, and D. Peleg. The dense k-subgraph problem.
*Algorithmica*, to appear.Google Scholar - 7.U. Feige and M. Seltser. On the densest k-subgraph problems. Technical ReportCS97-16, http://www.wisdom.weizmann.ac.il, 1997.
- 8.R. Hassin, S. Rubinstein, and A. Tamir. Approximation algorithms for maximum dispersion.
*Operations Research Letters*, 21:133–137, 1997.MATHCrossRefMathSciNetGoogle Scholar - 9.J. Håstad. Clique is hard to approximate within n
^{1-ε}. In*Proceedings of the 37th IEEE Annual Symposium on Foundations of Computer Science*, pages 627–636, 1996.Google Scholar - 10.G. Kortsarz and D. Peleg. On choosing a dense subgraph. In
*Proceedings of the 34th IEEE Annual Symposium on Foundations of Computer Science*, pages 692–701, Palo Alto, CA, 1993.Google Scholar - 11.F. P. Preparata and M. I. Shamos.
*Computational Geometry: An Introduction*. Springer-Verlag, New York, NY, 1985.Google Scholar - 12.S. S. Ravi, D. J. Rosenkrantz, and G. K. Tayi. Heuristic and special case algorithms for dispersion problems.
*Operations Research*, 42:299–310, 1994.MATHGoogle Scholar - 13.A. Srivastav and K. Wolf. Finding dense subgraphs with semidefinite programming. In K. Jansen and J. Rolim, editors,
*Approximation Algorithms for Combinatorial Optimization (APPROX’ 98)*, volume 1444 of*Lecture Notes in Computer Science*, pages 181–191, Aalborg, Denmark, 1998. Springer-Verlag.CrossRefGoogle Scholar - 14.A. Tamir. Obnoxious facility location in graphs.
*SIAM Journal on Discrete Mathematics*, 4:550–567, 1991.MATHCrossRefMathSciNetGoogle Scholar - 15.A. Tamir. Comments on the paper: ‘Heuristic and special case algorithms for dispersion problems’ by, S. S. Ravi, D. J. Rosenkrantz, and G. K. Tayi.
*Operations Research*, 46:157–158, 1998.CrossRefGoogle Scholar