Small approximate pareto sets for biobjective shortest paths and other problems

SIAM Journal on Computing

Volumn 39, Issue 4, 2009, Pages 1340-1371

  Diakonikolas, Ilias ^a   Yannakakis, Mihalis ^a  

^a Columbia University ^*  (United States)

Author keywords

Approximate Pareto set; Biobjective shortest path; Multiobjective optimization

Indexed keywords

APPROXIMATE PARETO SET; BIOBJECTIVE SHORTEST PATH; DUAL PROBLEM; MULTI-OBJECTIVE OPTIMIZATION PROBLEM; NP-HARD; PARETO CURVE; PARETO SET; POLYNOMIAL-TIME; SHORTEST PATH; SPANNING TREE; UPPER AND LOWER BOUNDS;

COMPUTATIONAL COMPLEXITY; COMPUTER SCIENCE; CONTROL THEORY; CURVE FITTING; PARETO PRINCIPLE; POLYNOMIAL APPROXIMATION; SET THEORY; WIRELESS SENSOR NETWORKS;

MULTIOBJECTIVE OPTIMIZATION;

EID: 70350180834     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/080724514     Document Type: Article

References (50)

1. E. Angel, E. Bampis, and A. Kononov, An FPTAS for approximating the unrelated parallel machines scheduling problem with costs, in Proceedings of the 9th Annual European Symposium on Algorithms, 2001, pp. 194-205.
2. E. Angel, E. Bampis, and A. Kononov, On the approximate trade-off for bicriteria batching and parallel machine scheduling problems, Theoret. Comput. Sci., 306 (2003), pp. 319-338.
3. H. Aissi, C. Bazgan, and D. Vanderpooten, Approximation complexity of min-max (regret) versions of shortest path, spanning tree and knapsack, in Proceedings of the 13th Annual European Symposium on Algorithms, 2005, pp. 862-873.
4. H. Aissi, C. Bazgan, and D. Vanderpooten, Complexity of min-max (regret) versions of cut problems, in Proceedings of the 16th Annual International Symposium on Algorithms and Computation, 2005, pp. 789-798. (Pubitemid 43856305)
5. H. Ackermann, A. Newman, H. Röglin, and B. Vöcking, Decision making based on approximate and smoothed Pareto curves, Theoret. Comput. Sci., 378 (2007), pp. 253-270.
6. A. F. Archer, Two O(log k)-approximation algorithms for the asymmetric k-center problem, in Proceedings of the 8th International Conference on Integer Programming and Combinatorial Optimization, 2001, pp. 1-14.
7. A. F. Archer, Private communication, 2007.
8. P. Assouad, Densité et Dimension, Ann. Inst. Fourier (Grenoble), 33 (1983), pp. 233-282.
9. A. Armon and U. Zwick, Multicriteria global minimum cuts, Algorithmica, 46 (2006), pp. 15-26.
10. A. Berger, V. Bonifaci, F. Grandoni, and G. Schäfer, Budgeted matching and budgeted matroid intersection via the gasoline puzzle, in Proceedings of the 13th International Conference on Integer Programming and Combinatorial Optimization, 2008, pp. 273-287.
11. H. Brönnimann and M. T. Goodrich, Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom., 14 (1995), pp. 463-479.
12. J. Chuzhoy, S. Guha, E. Halperin, S. Khanna, G. Kortsartz, R. Krauthgamer, and S. Naor, Asymmetric k-center is log n-hard to approximate, J. ACM, 52 (2005), pp. 538-551.
13. V. Chvátal, A greedy heuristic for the set covering problem, Math. Oper. Res., 4 (1979), pp. 233-235.
14. T. C. E. Cheng, A. Janiak, and M. Y. Kovalyov, Bicriterion single machine scheduling with resource dependent processing times, SIAM J. Optim., 8 (1998), pp. 617-630.
15. J. Climacao, ed., Multicriteria Analysis, Springer-Verlag, New York, 1997.
16. G. Chen and G. Xue, A PTAS for weight constrained Steiner trees in series-parallel graphs, Theoret. Comput. Sci., 304 (2003), pp. 237-247.
17. G. Chen and G. Xue, k-pair delay constrained minimum cost routing in undirected net- works, in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), ACM, New York, SIAM, Philadelphia, 2001, pp. 230-231.
18. K. L. Clarkson and K. Varadarajan, Improved approximation algorithms for geometric set cover, Discrete Comput. Geom., 37 (2007), pp. 43-58.
19. J. Dongarra, E. Jeannot, E. Saule, and Z. Shi, Bi-objective scheduling algorithms for optimizing makespan and reliability on heterogeneous systems, in Proceedings of the 19th Annual ACM Symposium on Parallel Algorithms and Architectures, 2007, pp. 280-288.
20. M. Ehrgott and X. Gandibleux, An annotated bibliography of multiobjective combinatorial optimization problems, OR Spectrum, 42 (2000), pp. 425-460.
21. M. Ehrgott, Multicriteria Optimization, 2nd ed., Springer-Verlag, Berlin, 2005.
22. G. Even, D. Rawitz, and S. Shahar, Hitting sets when the VC-dimension is small, Inform. Process. Lett., 95 (2005), pp. 358-362.
23. F. Ergun, R. Sinha, and L. Zhang, An improved FPTAS for restricted shortest path, Inform. Process. Lett., 83 (2002), pp. 237-239.
24. E. Erkut and V. Verter, Modeling of transport risk for hazardous materials, Oper. Res., 46 (1998), pp. 625-642.
25. U. Feige, A threshold of ln n for approximating set cover, J. ACM, 45 (1998), pp. 634-652.
26. J. Figueira, S. Greco, and M. Ehrgott, eds., Multiple Criteria Decision Analysis: State of the Art Surveys, Springer-Verlag, New York, 2005.
27. M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman, San Francisco, CA, 1979.
28. M. X. Goemans and R. Ravi, The constrained minimum spanning tree problem, in Algorithm Theory-SWAT '96: 5th Scandinavian Workshop on Algorithm Theory, Springer- Verlag, London, 1996, pp. 66-75.
29. A. Goel, K. G. Ramakrishnan, D. Kataria, and D. Logothetis, efficient computation of delay-sensitive routes from one source to all destinations, in Proceedings of the 12th Annual Joint Conference of the IEEE Computer and Communications Societies, 2001, pp. 854-858.
30. P. Hansen, Bicriterion path problems, in Proceedings of the 3rd Conference Multiple Criteria Decision Making Theory and Application, Lecture Notes in Econom. and Math. Systems 177, Springer-Verlag, Berlin, 1979, pp. 109-127.
31. R. Hassin, Approximation schemes for the restricted shortest path problem, Math. Oper. Res., 17 (1992), pp. 36-42.
32. R. Hassin and A. Levin, An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection, SIAM J. Comput., 33 (2004), pp. 261-268.
33. D. Haussler and E. Welzl, Epsilon-nets and simplex range queries, Discrete Comput. Geom., 2 (1987), pp. 127-151.
34. D. S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. System Sci., 9 (1978), pp. 256-278.
35. V. Koltun and C. H. Papadimitirou, Approximately dominating representatives, Theoret. Comput. Sci., 371 (2007), pp. 148-154.
36. J. Komlos, J. Pach, and W. Woeginger, Almost tight bounds for epsilon-nets, Discrete Comput. Geom., 7 (1992), pp. 10-15.
37. L. Lovász, On the ratio of optimal integral and fractional covers, Discrete Math., 13 (1975), pp. 383-390.
38. D. H. Lorenz and D. Raz, A simple efficient approximation scheme for the restricted shortest path problem, Oper. Res. Lett., 28 (2001), pp. 213-219.
39. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, J. ACM, 41 (1994), pp. 960-981.
40. K. M. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Boston, MA, 1999.
41. E. Pyrga and S. Ray, New existence proofs for-nets, in Proceedings of the 24th Annual ACM Symposium on Computational Geometry, 2008, pp. 199-207.
42. R. Panigrahy and S. Vishwanathan, An O(log n) approximation algorithm for the asymmetric p-center problem, J. Algorithms, 27 (1998), pp. 259-268.
43. J. Pach and W. Woeginger, Some new bounds for epsilon-nets, in Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, pp. 10-15.
44. C. H. Papadimitriou and M. Yannakakis, On the approximability of trade-offs and optimal access of web sources, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000, pp. 86-92.
45. C. H. Papadimitriou and M. Yannakakis, Multiobjective query optimization, in Proceedings of the 20th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, 2001, pp. 52-59.
46. G. Tsaggouris and C. D. Zaroliagis, Multiobjective optimization: Improved FPTAS for shortest paths and nonlinear objectives with applications, Theory Comput. Syst., 45 (2009), pp. 162-186.
47. V. N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl., 16 (1971), pp. 264-280.
48. P. Van Mieghen and L. Vandenberghe, Trade-off curves for QoS routing, in Proceedings of the 25th IEEE International Conference on Computer Communications, 2006, pp. 1-11.
49. S. Vassilvitskii and M. Yannakakis, effciently computing succinct trade-off curves, Theoret. Comput. Sci., 348 (2005), pp. 334-356.
50. A. Warburton, Approximation of Pareto optima in multiple-objective shortest path problems, Oper. Res., 35 (1987), pp. 70-79.