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Abstract

We consider the problem of computing minimum congestion, fault-tolerant, redundant assignments of messages to faulty parallel delivery channels. In particular, we are given a set M of faulty channels, each having an integer capacity c i and failing independently with probability f i . We are also given a set of messages to be delivered over M, and a fault-tolerance constraint (1– ε), and we seek a redundant assignment φ that minimize congestion Cong(φ), i.e. the maximum channel load, subject to the constraint that, with probability no less than (1– ε), all the messages have a copy on at least one active channel. We present a 4-approximation algorithm for identical capacity channels and arbitrary messages sizes, and a \(2 \left \lceil \frac{\ln(\vert M\vert/\epsilon)}{\ln(1/f_{\mathrm{max}})} \right \rceil\)-approximation algorithm for related capacity channels and unit size messages.

Both algorithms are based on computing a collection of disjoint channel subsets such that, with probability no less than (1– ε), at least one channel is active in each subset. The objective is to maximize the sum of the minimum subset capacities. Since the exact version of this problem is \(\mathcal{NP}\)-complete, we present a 2-approximation algorithm for identical capacities, and a (8 + o(1))-approximation algorithm for arbitrary capacities.

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References

  1. 1.
    Assmann, S.F., Johnson, D.S., Kleitman, D.J., Leung, J.Y.-T.: On a Dual Version of the One-Dimensional Bin Packing Problem. Journal of Algorithms 5, 502–525 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Fotakis, D.A., Spirakis, P.G.: Minimum Congestion Redundant Assignments to Tolerate Random Faults (1999), http://students.ceid.upatras.gr/~fotakis
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the The- ory of NP-Completeness. Freeman, San Francisco (1979)Google Scholar
  4. 4.
    Gasieniec, L., Kranakis, E., Krizanc, D., Pelc, A.: Minimizing Congestion of Layouts for ATM Networks with Faulty Links. In: Proc. of the 21st Mathematical Foundations of Computer Science, pp. 372–381 (1996)Google Scholar
  5. 5.
    Hochbaum, D.S. (ed.): Approximation Algorithms for NP-hard problems. PWS Publishing (1997)Google Scholar
  6. 6.
    Ibarra, O.H., Kim, C.E.: Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems. Journal of the Association for Computing Machinery 22, 463–468 (1975)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Kalyanasundaram, B., Pruhs, K.R.: Fault-Tolerant Scheduling. In: Proc. of the 26th ACM Symposium on Theory of Computing, pp. 115–124 (1994)Google Scholar
  8. 8.
    Karger, D.R.: A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem. In: Proc. of the 27th ACM Symposium on Theory of Computing, pp. 11–17 (1995)Google Scholar
  9. 9.
    Kleinberg, J., Rabani, Y., Tardos, E.: Allocating Bandwidth for Bursty Connections. In: Proc. of the 29th ACM Symposium on Theory of Computing, pp. 664–673 (1997)Google Scholar
  10. 10.
    Lomonosov, M.V.: Bernoulli Scheme with Closure. Problems of Information Transmission 10, 73–81 (1974)MathSciNetGoogle Scholar
  11. 11.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  12. 12.
    Toda, S., Watanabe, O.: Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science 100, 205–221 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Valiant, L.G.: The Complexity of Enumeration and Reliability Problems. SIAM Journal on Computing 8(3), 410–421 (1979)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Dimitris A. Fotakis
    • 1
    • 2
  • Paul G. Spirakis
    • 1
    • 2
  1. 1.Computer Engineering and Informatics DepartmentUniversity of PatrasRion, PatrasGreece
  2. 2.Computer Technology InstituteCTIPatrasGreece

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