Randomized rounding: A technique for provably good algorithms and algorithmic proofs

Abstract

We study the relation between a class of 0–1 integer linear programs and their rational relaxations. We give a randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem. Our technique can be a of extended to provide bounds on the disparity between the rational and 0–1 optima for a given problem instance.

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References

  1. [1]

    R. Aharoni, P. Erdős andN. Linial, Dual Integer Linear Programs and the Relationship between their Optima,Proceedings of the Seventeenth ACM Symposium on Theory of Computing, ACM, New York, May1985, 476–483.

    Google Scholar 

  2. [2]

    D. Angluin andL. G. Valiant, Fast probabilistic algorithms for Hamiltonian circuits and matchings,Journal of Computer and System Sciences,19, 155–193.

  3. [3]

    H. Chernoff, A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations,Annals of Math. Stat.,23 (1952), 493–509.

    MathSciNet  Google Scholar 

  4. [4]

    S. Even, A. Itai andA. Shamir, On the complexity of timetable and multi-commodity flow problems,SIAM Journal on Computing,5 (1976), 691–703.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs,Combinatorica 1 (1981), 155–162.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    W. Hoeffding, On the distribution of the number of successes independent trials,Annals of Math. Stat.,27 (1956), 713–721.

    MathSciNet  Google Scholar 

  7. [7]

    T. C. Hu andM. T. Shing, A Decomposition Algorithm for Circuit Routing,Mathematical Programming Study,24 (1985), 87–103.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    N. Karmarkar, A new polynomial-time algorithm for linear programming,Combinatorica 4 (1984), 373–396.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    R. M. Karp, Reducibility among combinatorial problems,Complexity of Computer Computations, (ed. R. N. Miller, J. W. Thatcher),Plenum Press, New York, (1972), 85–104.

    Google Scholar 

  10. [10]

    R. M. Karp, F. T. Leighton, R. L. Rivest, C. D. Thompson, U. Vazirani andV. Vazirani Global Wire Routing in Two-Dimensional Arrays,Proc. 24th Annual Symp. on Foundations of Computer Science, (1983), 453–459.

  11. [11]

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

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    V. M. Malhotra, M. P. Kumar andS. N. Maheshwari, AnO(|V|3) algorithm for finding maximum flows in networks,Information Processing Letters,7 (1978), 277–278.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    P. Raghavan andC. D. Thompson, Randomized Routing in Gate-Arrays,CSD/84/202, Computer Science Division, UC Berkeley, (1984).

  14. [14]

    P. Raghavan andC. D. Thompson, Provably Good Routing in Graphs: Regular Arrays,Proceedings of the Seventeenth ACM Symposium on Theory of Computing, ACM, New York, (1985), 79–87.

  15. [15]

    P. Raghavan, Randomized Rouding and Discrete Ham-Sandwiches: Provably Good Algorithms for Routing and Packing Problems,PhD Thesis, Computer Science Division, UC Berkeley, (1986).

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This work was supported by Semiconductor Research Corporation grant SRC 82-11-008 and an IBM Doctoral Fellowship.

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Raghavan, P., Tompson, C.D. Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica 7, 365–374 (1987). https://doi.org/10.1007/BF02579324

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  • 90 C 10