Algorithmic Chernoff-Hoeffding inequalities in integer programming
Are there more examples of NP-hard combinatorial optimization problems for which derandomization yields constant factor approximations in polynomial-time ?
The pessimistic estimator technique shows an O(mn)-time implementation of the conditional probability method on the RAM model of computation in case of m large deviation events associated to m unweighted sums of n indepependent Bernoulli trials. Is there a fast algorithm also in case of rational weighted sums of Bernoulli trials ?
This paper is a contribution to both problems. On the one hand we will present the first implementation of the conditional probability method on the RAM model of computation for rational weighted sums of Bernoulli trials, which needs O(mn2 log mn/ε)-time, where ε is the success probability of the given m events. Hence the present gap between the running times of the unweighted and the weighted conditional probability method is a O(n log mn/ε)-factor.
On the other hand we will show, applying such derandomization procedures, which we call the algorithmic Chernoff-Hoeffding inequalities, the first constant factor approximation algorithms for maximizing integer programs with rational constraint matrix and integer domains, provided that the entries of the constraint vector grow logarithmically in the number of constraints.
Unable to display preview. Download preview PDF.
- N. Alon, J. Spencer, P. Erdös; The probabilistic method. John Wiley & Sons, Inc. 1992.Google Scholar
- H. Chernoff; A measure of asymptotic efficiency for test of a hypothesis based on the sum of observation. Ann. Math. Stat. 23, (1952), 493–509.Google Scholar
- A. Feldstein, P.R. Turner; Overflow, underflow, and severe loos of significance in floating point addition and subtraction. IMA J. Numer. Anal ysis, Vol. 6, (1986), 241–251.Google Scholar
- M. R. Garey, D. S. Johnson; Computers and Intractability. W. H. Freeman and Company, New York (1979).Google Scholar
- M. Grötschel, L. Lovász, A. Schrijver; Geometric algorithms and combinatorial optimixation. Springer-Verlag (1988).Google Scholar
- W. Hoeffding; On the distribution of the number of success in independent trials. Annals of Math. Stat. 27, (1956), 713–721.Google Scholar
- J. K. Lenstra, D. B. Shmoys, E. Tardos; Approximating algorithms for scheduling unrelated parallel machines. Proceedings of the 28.th Annual IEEE Symposium on the Foundations of Computer Science (1987).Google Scholar
- C. McDiarmid; On the method of bounded differences. Surveys in Combinatorics, 1989. J. Siemons, Ed.: London Math. Soc. Lectures Notes, Series 141, Cambridge University Press, Cambridge, England 1989.Google Scholar
- P. Raghavan, C. D. Thompson; Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (4), (1987), 365–374.Google Scholar
- P. Raghavan; Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Jour. of Computer and System Sciences 37, (1988), 130–143.Google Scholar
- A. Srivastav, P. Stangier; Integer multicommodity flows with reduced demands. in T. Lengauer (eds.), Proceedings of the First Annual European Symposium on Algorithms (ESA '93), Bonn 1993, Lecture Notes in Computer Science 726, pp. 360–372, Springer Verlag (1993).Google Scholar
- A. Srivastav, P. Stangier; Weighted fractional and integral k-matching in hypergraphs. To appear in Disc.Appl.Math. (1994).Google Scholar
- A. Srivastav, P. Stangier; On quadratic lattice approximations. in Proceedings of the Fourth Annual International Symposium on Algorithms and Computations, ISAAC'93, Hong-Kong, Lecture Notes in Computer Science 762, pp.176–184, Springer Verlag (1993).Google Scholar
- C-K. Yap; Towards exact geometric computation. Preprint, Courant Institute, (1993).Google Scholar