Abstract
In this paper, we give the first polynomial time approximation algorithms for two problems in combinatorial optimization. The first problem is single-processor scheduling to minimize weighted sum of completion times, subject to precedence constraints. The second problem, interval graph completion, is finding a minimum-size interval graph containing the input graph as a subgraph. Both problems are NP-complete; our algorithms output solutions that are within a polylogarithmic factor of optimal. To achieve these bounds, we make use of a technique developed and first applied by Leighton and Rao [12], together with a technique of Hansen [5].
Research supported by NSF grant CCR-9012357, NSF grant CDA 8722809, ONR and DARPA contract N00014-83-K-0146 and ARPA Order No. 6320, Amendment 1.
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References
D. Adolphson and T. C. Hu, “Optimal linear ordering”, SIAM J. Appl. Math. 25 (1973), pp. 403–423.
R. W. Conway, W. L. Maxwell, and L. W. Miller, Theory of Scheduling (1967), Addison-Wesley, Reading, Massachusetts.
M. R. Garey and D. S. Johnson, Computers and Intractability: A guide to the theory of NP-completeness, W. H. Freeman, San Francisco (1979).
M. R. Garey., D. S. Johnson, and L. Stockmeyer, “Some simplified NP-complete graph problems,” Theor. Comput. Sci. 1 (1976), pp. 237–267.
Mark D. Hansen, “Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems,” Proceedings, 30th Symposium on Foundations of Computer Science (1989), pp. 604–609.
W. A. Horn, “Single machine job sequencing with treelike precedence ordering and linear delay penalties,” SIAM J. Appl. Math. 23 (1972), pp. 189–202.
D. G. Kendall, “Incidence matrices, interval graphs, and seriation in archeology,” Pacific J. Math. 28 (1969), pp. 565–570.
P. Klein, A. Agrawal, R. Ravi, and S. Rao, “Approximation through multicommodity flow”, Proceedings, 31st Annual Symp. on Foundations of Comp. Sci., (1990), pp. 726–737.
E. L. Lawler, “Sequencing jobs to minimize total weighted completion time subject to precedence constraints,” Annals of Discrete Math. 2 (1978), pp. 75–90.
F. T. Leighton, F. Makedon, S. Plotkin, C. Stein, E. Tardos, S. Tragoudas, “Fast approximation algorithms for multicommodity flow problems,” Proceedings, 23rd Annual ACM Symposium on Theory of Computing (1991), to appear.
F. T. Leighton, F. Makedon, and S. Tragoudas, personal communication, 1990
F. T. Leighton and S. Rao, “An approximate max-flow min-cut theorem for uniform multicommodity flow problems with application to approximation algorithms”, Proceedings, 29th Symposium on Foundations of Computer Science (1988), pp. 422–431.
G. Ramalingam, and C. Pandu Rangan, “A unified approach to domination problems in interval graphs”, Information Processing Letters, vol. 27 (1988), pp. 271–274.
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© 1991 Springer-Verlag Berlin Heidelberg
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Ravi, R., Agrawal, A., Klein, P. (1991). Ordering problems approximated: single-processor scheduling and interval graph completion. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds) Automata, Languages and Programming. ICALP 1991. Lecture Notes in Computer Science, vol 510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54233-7_180
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DOI: https://doi.org/10.1007/3-540-54233-7_180
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