Abstract
The MAX-MIN dispersion problem, which arises in the placement of undesirable facilities, involves selecting a specified number of sites among a set of potential sites so as to maximize the minimum distance between any pair of selected sites. We consider different versions of this dispersion problem where each potential site has an associated storage capacity and a storage cost. A typical problem in this context is to choose a subset of potential sites so that the total capacity of the chosen sites is at least a given value, the total storage cost is within the specified budget and the minimum distance between any pair of chosen sites is maximized. Since these constrained optimization problems are NP-hard in general, we consider whether there are efficient approximation algorithms for them with good performance guarantees. Our results include approximation algorithms for some versions, approximation schemes for some geometric versions and polynomial algorithms for special cases. We also present results that bring out the intrinsic difficulty of obtaining near-optimal solutions to some versions.
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References
Y. Asahiro, K. Iwama, H. Tamaki, and T. Tokuyama, “Greedily Finding a Dense Subgraph,” in Proc. Scandinavian Workshop on Algorithm Theory (SWAT'96), Reykjavik, Iceland, July 1996, pp. 136-148.
C. Baur and S.P. Fekete, “Approximation of Geometric Dispersion Problems,” in Proc. First International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'98), Aalborg, Denmark, July 1998, pp. 63-75 (Springer Lecture Notes in Computer Science, vol. 1444).
M. Bellare and M. Sudan, “Improved Non-approximability Results,” in Proc. 26th Annual ACM Symposium on Theory of Computing (STOC'94), Montreal, Canada, May 1994, pp. 184-193.
H. Breu and D.G. Kirkpatrick, “Unit disk graph recognition is NP-hard,” Computational Geometry, vol. 9, pp. 3-24, 1988.
B. Chandra and M.M. Halldórsson, “Facility Dispersion and Remote Subgraphs,” Proc. Scandinavian Workshop on Algorithm Theory (SWAT'96), Reykjavik, Iceland, July 1996.
R. Chandrasekharan and A. Daughety, “Location on tree networks: p-centre and n-dispersion problems,” Mathematics of Operations Research, vol. 6, no. 1, pp. 50-57, 1981.
R.L. Church and R.S. Garfinkel, “Locating an obnoxious facility on a network,” Transportation Science, vol. 12, no. 2, pp. 107-118, 1978.
B.N. Clark, C.J. Colbourn, and D.S. Johnson, “Unit disk graphs,” Discrete Mathematics, vol. 86, pp. 165-177, 1990.
T. Cormen, C.E. Leiserson, and R. Rivest, Introduction to Algorithms, MIT Press/McGraw-Hill: Cambridge, MA, 1991.
E. Erkut and S. Neuman, “Analytical models for locating undesirable facilities,” European Journal of Operational Research, vol. 40, pp. 275-291, 1989.
E. Erkut and S. Neuman, “Comparison of four models for dispersing facilities,” INFOR, vol. 29, pp. 68-85, 1990.
E. Erkut, C. ReVelle, and Y. Ulkusal, “Integer-friendly formulations for the r-separation problem,” European Journal of Operational Research, vol. 92, pp. 342-351, 1996.
U. Feige, G. Kortsarz, and D. Peleg, “The dense k-subgraph problem,” Private communication, April 1999.
G.N. Frederickson and D.B. Johnson, “The complexity of selection and ranking in X +Y and matrices with sorted columns,” Journal of Computer and System Sciences, vol. 24, no. 2, pp. 197-208, 1982.
G.N. Frederickson and D.B. Johnson, “Generalized selection and ranking: Sorted matrices,” SIAM Journal on Computing, vol. 13, no. 1, pp. 14-30, 1984.
Z. Galil and K. Park, “A linear time algorithm for concave one-dimensional dynamic programming,” Information Processing Letters, vol. 33, no. 6, pp. 309-311, 1990.
Z. Galil and K. Park, “Dynamic programming with convexity, concavity and sparsity,” Theoretical Computer Science, vol. 92, no. 1, pp. 49-76, 1992.
M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Co.: San Francisco, CA, 1979.
A.J. Goldman and P.M. Dearing, “Concepts of optimal locations for partially noxious facilities,” Bulletin of the Operational Research Society of America, vol. 23, no. 1, B85, 1975.
M.M. Halldórsson, K. Iwano, N. Katoh, and T. Tokuyama, “Finding Subsets Maximizing Minimum Structures,” in Proc. Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'95), San Francisco, CA, Jan. 1995, pp. 150-157.
P. Hansen and I.D. Moon, “Dispersing facilities on a network,” Presentation at the TIMS/ORSA Joint National Meeting, Washington, DC, 1988.
R. Hassin and A. Tamir, “Improved complexity bounds for location problems on the real line,” Operations Research Letters, vol. 10, pp. 395-402, 1991.
R. Hassin, S. Rubinstein, and A. Tamir, “Approximation algorithms for maximum dispersion,” Operations Research Letters, vol. 21, pp. 133-137, 1997.
J. Håstad, “Clique is Hard to Approximate Within n 1-∈,” in Proc. 37th Annual Symposium on Foundations of Computer Science (FOCS'96), Burlington, VT, pp. 627-626, 1996.
D.S. Hochbaum and W. Maass, “Approximation schemes for covering and packing problems in image processing and VLSI,” J. ACM, vol. 32, no. 1, pp. 130-136, 1985.
H.B. Hunt III, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz, and R.E. Stearns, “NC-approximation schemes for NP-and PSPACE-hard problems for geometric graphs,” Journal of Algorithms, vol. 26, no. 2, pp. 238-274, 1998.
G. Kortsarz and D. Peleg, “On Choosing a Dense Subgraph,” in Proc. 34th Symposium on Foundations of Computer Science (FOCS'93), Palo Alto, CA, pp. 692-701, 1993.
M.V. Marathe, H. Breu, H.B. Hunt III, S.S. Ravi, and D.J. Rosenkrantz, “Simple heuristics for unit disk graphs,” Networks, vol. 25, pp. 59-68, 1995.
S. Martello and P. Toth, Knapsack Problems: Algorithms and Computer Implementations, John Wiley & Sons: New York, NY, 1990.
E. Melachrinoudis and T.P. Cullinane, “Locating an undesirable facility with a minimax criterion,” European Journal of Operational Research, vol. 24, pp. 239-246, 1986.
P.B. Mirchandani and R.L. Francis, Discrete Location Theory, John Wiley and Sons, Inc: New York, NY, 1990.
I.D. Moon and S.S. Chaudhry, “An analysis of network location problems with distance constraints,” Management Science, vol. 30, no. 3, pp. 290-307, 1984.
C.H. Papadimitriou, Computational Complexity, Addison-Wesley Publishing Co.: Reading, MA, 1994.
J. Paroush and C.S. Tapiero, “Optimal location of a polluting plant on a line under uncertainty,” Journal of Regional Science, vol. 16, no. 3, pp. 365-374, 1976.
S.S. Ravi, D.J. Rosenkrantz, and G.K. Tayi, “Heuristic and special case algorithms for dispersion problems,” Operations Research, vol. 42, no. 2, pp. 299-310, 1994.
D.J. Rosenkrantz, G.K. Tayi, and S.S. Ravi, “Capacitated Facility Dispersion Problems,” Technical Report, Department of Computer Science, University at Albany-State University of New York, Aug. 1998.
A. Tamir, “Obnoxious facility location on graphs,” SIAM Journal on Discrete Mathematics, vol. 4, pp. 550-567, 1991.
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, vol. 46, no. 1, pp. 157-158, 1998.
D.J. White, “The maximal dispersion problem and the 'First point outside the neighborhood' heuristic,” Computers in Operations Research, vol. 18, no. 1, pp. 43-50, 1991.
D.W. Wong and Y.S. Kuo, “A study of two geometric location problems,” Information Processing Letters, vol. 28, no. 6, pp. 281-286, 1988.
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Rosenkrantz, D.J., Tayi, G.K. & Ravi, S. Facility Dispersion Problems Under Capacity and Cost Constraints. Journal of Combinatorial Optimization 4, 7–33 (2000). https://doi.org/10.1023/A:1009802105661
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DOI: https://doi.org/10.1023/A:1009802105661