Abstract
We consider the problem of placing a specified number p of facilities on the nodes of a given network with two nonnegative edge-weight functions so as to minimize the diameter of the placement with respect to the first weight function subject to a diameter- or sumconstraint with respect to the second weight function.
Define an (α, β)-approximation algorithm as a polynomial-time algorithm that produces a solution within α times the optimal value with respect to the first weight function, violating the constraint with respect to the second weight function by a factor of at most β.
We show that in general obtaining an (α,β)-approximation for any fixed α, β≥1 is NP-hard for any of these problems. We also present efficient approximation algorithms for several of the problems studied, when both edge-weight functions obey the triangle inequality.
Research supported by Department of Energy under contract W-7405-ENG-36.
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References
B. Awerbuch, A. Baratz, and D. Peleg. Cost-sensitive analysis of communication protocols. In Proceedings of the 9th Symposium on the Principles of Distributed Computing (PODC), pages 177–187, 1990.
E.M. Arkin, S.P. Fekete, J.S.B. Mitchell, and C.D. Piatko. Optimal Covering Tour Problems. In Proceedings of the Fifth International Symposium on Algorithms and Computation, 1994.
A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with Minimum Diameter and Related Problems. J. Algorithms, 12(1):38–56, March 1991.
J. Bar-Ilan and D. Peleg. Approximation Algorithms for Selecting Network Centers. In Proc. 2nd Workshop on Algorithms and Data Structures (WADS), pages 343–354, Ottawa, Canada, August 1991. Springer Verlag, LNCS vol. 519.
M. Bellare and M. Sudan. Improved Non-Approximability Results. in Proceedings of the 26th annual ACM Symposium on the Theory of Computing (STOC), May 1994.
M.E. Dyer and A.M. Frieze. A Simple Heuristic for the p-Center Problem. Operations Research Letters, 3(6):285–288, Feb. 1985.
E. Erkut and S. Neuman. Analytical Models for Locating Undesirable Facilities. European J. Operations Research, 40:275–291, 1989.
T. Feder and D. Greene. Optimal Algorithms for Approximate Clustering. In ACM Symposium on Theory of Computing (STOC), pages 434–444, 1988.
T. Fischer. Optimizing the Degree of Minimum Weight Spanning Trees. Technical report, Department of Computer Science, Cornell University, Ithaca, New York, April 1993.
M.R. Garey and D.S. Johnson. Computers and Intractability. W.H. Freeman, 1979.
T.F. Gonzalez. Clustering to Minimize the Maximum Intercluster Distance. Theoretical Computer Science, 38:293–306, 1985.
R. Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research, 17(1):36–42, 1992.
G.Y. Handler and P.B. Mirchandani. Location on Networks: Theory and Algorithms. MIT Press, Cambridge, MA, 1979.
D. S. Hochbaum and D. B. Shmoys. A Unified Approach to Approximation Algorithms for Bottleneck Problems. Journal of the ACM, 33(3):533–550, July 1986.
A. Iwainsky, E. Canuto, O. Taraszow, and A. Villa. Network decomposition for the optimization of connection structures. Networks, 16:205–235, 1986.
S. Khuller, B. Raghavachari, and N. Young. Balancing Minimum Spanning and Shortest Path Trees. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 243–250, 1993.
D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478–487, 1982.
J.H. Lin and J. S. Vitter. ε-Approximations with Minimum Packing Constraint Violation. In ACM Symposium on Theory of Computing (STOC), pages 771–781, May 1992.
M.V. Marathe, R. Ravi, R. Sundaram, S.S. Ravi, D.J. Rosenkrantz, and H.B. Hunt III. Bicriteria Network Design Problems. To appear in Proceedings of the 22nd International Colloquium on Automata Languages and Programming (ICALP), 1995.
P.B. Mirchandani and R.L. Francis. Discrete Location Theory. Wiley-Interscience, New York, NY, 1990.
F.P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Springer-Verlag Inc., New York, NY, 1985.
V. Radhakrishnan, S.O. Krumke, M.V. Marathe, D.J. Rosenkrantz, and S.S. Ravi. Compact Location Problems. In 13th Conference on the Foundations of Software Technology and Theoretical Computer Science (FST-TCS), volume 761 of LNCS, pages 238–247, December 1993.
R. Ravi, M.V. Marathe, S.S. Ravi, D.J. Rosenkrantz, and H.B. Hunt III. Many birds with one stone: Multi-objective approximation algorithms. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (STOC), pages 438–447, 1993.
S.S. Ravi, D.J. Rosenkrantz, and G.K. Tayi. Facility Dispersion Problems: Heuristics and Special Cases. In Proc. 2nd Workshop on Algorithms and Data Structures (WADS), pages 355–366, Ottawa, Canada, August 1991. Springer Verlag, LNCS vol. 519. (Journal version: Operations Research, 42(2):299–310, March–April 1994.)
D.B. Shmoys and E. Tardos. Scheduling unrelated parallel machines with costs. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 438–447, 1993.
A. Warburton. Approximation of Pareto optima in multiple-objective, shortest path problems. Operations Research, 35:70–79, 1987.
R.T. Wong. Worst case analysis of network design problem heuristics. SIAM J. Disc. Math., 1:51–63, 1980.
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Krumke, S.O., Noltemeier, H., Ravi, S.S., Marathe, M.V. (1995). Compact location problems with budget and communication constraints. In: Du, DZ., Li, M. (eds) Computing and Combinatorics. COCOON 1995. Lecture Notes in Computer Science, vol 959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0030872
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DOI: https://doi.org/10.1007/BFb0030872
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