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The Local Ratio Technique and Its Application to Scheduling and Resource Allocation Problems

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Graph Theory, Combinatorics and Algorithms

Part of the book series: Operations Research/Computer Science Interfaces Series ((volume 34))

Abstract

We give a short survey of the local ratio technique for approximation algorithms, focusing mainly on scheduling and resource allocation problems.

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References

  1. A. Agrawal, P. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing, 24(3): 440–456 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Albers, S. Arora, and S. Khanna. Page replacement for general caching problems. In 10th Annual ACM-SIAM Symposium on Discrete Algorithms, (1999) pp. 31–40.

    Google Scholar 

  3. E. M. Arkin and E. B. Silverberg. Scheduling jobs with fixed start and end times. Discrete Applied Mathematics, 18: 1–8 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Arora and C. Lund. Hardness of approximations. In Hochbaum [44], chapter 10, pp. 399–446.

    Google Scholar 

  5. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45(3): 501–555 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1): 70–122 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation; Combinatorial optimization problems and their approximability properties. Springer Verlag, (1999).

    Google Scholar 

  8. M. Azizoglu and S. Webster. Scheduling a batch processing machine with incompatible job families. Computer and Industrial Engineering, 39(3–4): 325–335 (2001).

    Google Scholar 

  9. V. Bafna, P. Berman, and T. Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics, 12(3): 289–297 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Baptiste. Batching identical jobs. Mathematical Methods of Operations Research, 52(3): 355–367 (2000).

    MATH  MathSciNet  Google Scholar 

  11. A. Bar-Noy, R. Bar-Yehuda, A. Freund, J. Naor, and B. Shieber. A unified approach to approximating resource allocation and schedualing. Journal of the ACM, 48(5): 1069–1090 (2001).

    Article  MathSciNet  Google Scholar 

  12. A. Bar-Noy, S. Guha, Y. Katz, J. Naor, B. Schieber, and H. Shachnai. Throughput maximization of real-time scheduling with batching. In 13th Annual ACM-SIAM Symposium on Discrete Algorithms, (2002) pp. 742–751.

    Google Scholar 

  13. A. Bar-Noy, S. Guha, J. Naor, and B. Schieber. Approximating the throughput of multiple machines in real-time scheduling. SIAM Journal on Computing, 31(2): 331–352 (2001).

    MathSciNet  MATH  Google Scholar 

  14. Bar-Yehuda. Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms, 39(2): 137–144 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Bar-Yehuda. One for the price of two: A unified approach for approximating covering problems. Algorithmica, 27(2): 131–144 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2(2): 198–203 (1981).

    MathSciNet  MATH  Google Scholar 

  17. R. Bar-Yehuda and S. Even. A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25:27–46 (1985).

    MathSciNet  Google Scholar 

  18. R. Bar-Yehuda, D. Geiger, J. Naor, and R. M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference. SIAM Journal on Computing, 27(4): 942–959 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Bar-Yehuda and D. Rawitz. On the equivalence between the primal-dual schema and the local-ratio technique. In 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, volume 2129 of LNCS, (2001) pp. 24–35.

    MathSciNet  Google Scholar 

  20. A. Becker and D. Geiger. Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence, 83(1): 167–188 (1996).

    Article  MathSciNet  Google Scholar 

  21. P. Berman and B. DasGupta. Multi-phase algorithms for throughput maximization for real-time scheduling. Journal of Combinatorial Optimization, 4(3): 307–323 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Bertsimas and C. Teo. From valid inequalities to heuristics: A unified view of primaldual approximation algorithms in covering problems. Operations Research, 46(4): 503–514 (1998).

    MathSciNet  MATH  Google Scholar 

  23. R. Bhatia, J. Chuzhoy, A. Freund, and J. Naor. Algorithmic aspects of bandwidth trading. In 30th International Colloquium on Automata, Languages, and Programming, volume 2719 of LNCS, (2003) pp. 751–766.

    MathSciNet  Google Scholar 

  24. P. Brucker, A. Gladky, H. Hoogeveen, M. Y. Kovalyov, C. N. Potts, T. Tautenhahn, and S. L. van de Velde. Scheduling a batching machine. Journal of Scheduling, 1(1): 31–54 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  25. N. H. Bshouty and L. Burroughs. Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In 15th Annual Symposium on Theoretical Aspects of Computer Science, volume 1373 of LNCS, pp. 298–308. Springer, (1998).

    MathSciNet  Google Scholar 

  26. F. A. Chudak, M. X. Goemans, D. S. Hochbaum, and D. P. Williamson. A primal-dual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Operations Research Letters, 22: 111–118 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Chuzhoy, R. Ostrovsky, and Y. Rabani. Approximation algorithms for the job interval selection problem and related scheduling problems. In 42nd IEEE Symposium on Foundations of Computer Science, (2001) pp. 348–356.

    Google Scholar 

  28. V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3): 233–235 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  29. I. Dinur and S. Safra. The importance of being biased. In 34th ACM Symposium on the Theory of Computing, (2002) pp. 33–42.

    Google Scholar 

  30. G. Dobson and R. S. Nambimadom. The batch loading and scheduling problem. Operations Research, 49(1): 52–65 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Erdös and L. Pósa. On the maximal number of disjoint circuits of a graph. Publ. Math. Debrecen, 9: 3–12 (1962).

    MathSciNet  MATH  Google Scholar 

  32. U. Feige. A threshold of ln n for approximating set cover. In 28th Annual Symposium on the Theory of Computing, (1996) pp. 314–318.

    Google Scholar 

  33. T. Fujito. A unified approximation algorithm for node-deletion problems. Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science, 86: 213–231 (1998).

    MATH  MathSciNet  Google Scholar 

  34. R. Gandhi, S. Khuller, and A. Srinivasan. Approximation algorithms for partial covering problems. In 28th International Colloquium on Automata, Languages and Programming, volume 2076 of LNCS, (2001) pp. 225–236.

    MathSciNet  Google Scholar 

  35. M. R. Garey and D. S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, (1979).

    Google Scholar 

  36. M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified np-complete graph problems. Theoretical Computer Science, 1: 237–267 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  37. M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24(2): 296–317 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  38. M.X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In Hochbaum [44], chapter 4, pp. 144–191.

    Google Scholar 

  39. M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, (1980). Second edition: Annuals of Discrete Mathematics, 57, Elsevier, Amsterdam (2004).

    Google Scholar 

  40. E. Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In 11th Annual ACM-SIAM Symposium on Discrete Algorithms, (2000) pp. 329–337.

    Google Scholar 

  41. J. Håstad. Some optimal inapproximability results. In 29th Annual ACM Symposium on the Theory of Computing, (1997) pp. 1–10.

    Google Scholar 

  42. D. S. Hochbaum. Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing, 11(3): 555–556 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  43. D. S. Hochbaum. Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics, 6: 243–254 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  44. D. S. Hochbaum, editor. Approximation Algorithms for NP-Hard Problem. PWS Publishing Company, (1997).

    Google Scholar 

  45. K. Jansen. An approximation algorithm for the license and shift class design problem. European Journal of Operational Research, 73: 127–131 (1994).

    Article  MATH  Google Scholar 

  46. D. S. Johnson. Approximation algorithms for combinatorial problems. J. Comput. System Sci., 9: 256–278 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  47. R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J.W. Thatcher, editors, Complexity of Computer Computations, pp. 85–103, New York, (1972). Plenum Press.

    Google Scholar 

  48. M. Kearns. The Computational Complexity of Machine Learning. M.I.T. Press, (1990).

    Google Scholar 

  49. A. W. J. Kolen and L. G. Kroon. On the computational complexity of (maximum) class scheduling. European Journal of Operational Research, 54: 23–38 (1991).

    Article  MATH  Google Scholar 

  50. A. W. J. Kolen and L. G. Kroon. An analysis of shift class design problems. European Journal of Operational Research, 79: 417–430 (1994).

    Article  MATH  Google Scholar 

  51. J. M. Lewis and M. Yannakakis. The node-deletion problem for hereditary problems is NP-complete. Journal of Computer and System Sciences, 20: 219–230 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  52. L. Lovász. On the ratio of optimal integral and fractional covers. Discreate Mathematics, 13: 383–390 (1975).

    MATH  Google Scholar 

  53. C. Lund and M. Yannakakis. The approximation of maximum subgraph problems. In 20th International Colloquium on Automata, Languages and Programming, volume 700 of LNCS, July (1993) pp. 40–51.

    Google Scholar 

  54. S. V. Mehta and R. Uzsoy. Minimizing total tardiness on a batch processing machine with incompatable job famalies. IIE Transactions, 30(2): 165–178 (1998).

    Article  Google Scholar 

  55. B. Monien and R. Shultz. Four approximation algorithms for the feedback vertex set problem. In 7th conference on graph theoretic concepts of computer science, (1981) pp. 315–390.

    Google Scholar 

  56. B. Monien and E. Speckenmeyer. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica, 22: 115–123 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  57. G. L. Nemhauser and L. E. Trotter. Vertex packings: structural properties and algorithms. Mathematical Programming, 8: 232–248 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  58. R. Ravi and P. Klein. When cycles collapse: A general approximation technique for constrained two-connectivity problems. In 3rd Conference on Integer Programming and Combinatorial Optimization, (1993) pp. 39–56.

    Google Scholar 

  59. R. Raz and S. Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In 29th ACM Symposium on the Theory of Computing, (1997) pp. 475–484.

    Google Scholar 

  60. P. Slavík. Improved performance of the greedy algorithm for partial cover. Information Processing Letters, 64(5): 251–254 (1997).

    MathSciNet  Google Scholar 

  61. F. C. R. Spieksma. On the approximability of an interval scheduling problem. Journal of Scheduling, 2(5): 215–227 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  62. R. Uzsoy. Scheduling batch processing machines with incompatible job families. International Journal of Production Research, 33: 2685–2708 (1995).

    MATH  Google Scholar 

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Bar-Yehuda, R., Bendel, K., Freund, A., Rawitz, D. (2005). The Local Ratio Technique and Its Application to Scheduling and Resource Allocation Problems. In: Golumbic, M.C., Hartman, I.BA. (eds) Graph Theory, Combinatorics and Algorithms. Operations Research/Computer Science Interfaces Series, vol 34. Springer, Boston, MA. https://doi.org/10.1007/0-387-25036-0_5

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