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Probabilistic analysis of optimum partitioning

Published online by Cambridge University Press:  14 July 2016

Narendra Karmarkar ,
Richard M. Karp ,
George S. Lueker  and
Andrew M. Odlyzko
Narendra Karmarkar*
Affiliation:
AT&T Bell Laboratories
Richard M. Karp*
Affiliation:
University of California, Berkeley
George S. Lueker*
Affiliation:
University of California, Irvine
Andrew M. Odlyzko*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, Murray Hill, NJ 07974, USA.
∗∗Postal address: Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720, USA.
∗∗∗Postal address: Dept. of Information and Computer Science, University of California, Irvine, CA 92717, USA.
∗Postal address: AT&T Bell Laboratories, Murray Hill, NJ 07974, USA.
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Abstract

Given a set of n items with real-valued sizes, the optimum partition problem asks how it can be partitioned into two subsets so that the absolute value of the difference of the sums of the sizes over the two subsets is minimized. We present bounds on the probability distribution of this minimum under the assumption that the sizes are independent random variables drawn from a common distribution. For a large class of distributions, we determine the asymptotic behavior of the median of this minimum, and show that it is exponentially small.

Keywords

PARTITION PROBLEMSECOND-MOMENT METHODCOMBINATORIAL OPTIMIZATION
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 626 - 645
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

†

Supported by NSF Grant MCS 81-05217 at the University of California at Berkeley.

‡

Supported by NSF Grants MCS 79-04997 and DCR 85-09667 at the University of California at Irvine.

References

D'Atri, G. and Puech, C. (1982) Probabilistic analysis of the subset-sum problem. Discr. Appl. Math. 4, 329–334.Google Scholar
Billard, P. (1965) Séries de Fourier aléatoirement bornées, continues, uniformémente convergentes. Ann. Sci. Ec. Norm. Sup. 82, 131–179.Google Scholar
Boxma, O. J. (1984) Probabilistic analysis of the LPT schcduling rule. In Performance '84, ed. Gelenbe, E., North-Holland, Amsterdam, 475–490.Google Scholar
Bruno, J. L. and Downey, P. J. (1982) Probabilistic bounds on the performance of list scheduling. Technical Report TR 82-19, Department of Computer Science, The University of Arizona.Google Scholar
Coffman, E. G., Flatto, L. Jr. and Lueker, G. S. (1984) Expected makespans for largest-first multiprocessor scheduling. In Performance '84, ed. Gelenbe, E., North-Holland, Amsterdam, 491–506.Google Scholar
Coffman, E. G. Jr., Frederickson, G. N. and Lueker, G. S. (1984) A note on expected makespans for largest-first sequences of independent tasks on two processors. Math. Operat. Res. 9, 260–266.Google Scholar
Erdös, P. and Spencer, J. (1974) Probabilistic Methods in Combinatorics. Academic Press, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
Frenk, J. B. G., and Rinnooy Kan, A. H. G. (1983) The asymptotic optimality of the LPT heuristic. Econometrics Institute, Erasmus University, Rotterdam, The Netherlands.Google Scholar
Frenk, J. B. G. and Rinnooy Kan, A. H. G. (1984) On the rate of convergence to optimality of the LPT rule. Working Paper # 1610–84, Econometrics Institute, Erasmus University, Rotterdam, The Netherlands.Google Scholar
Garey, M. R. and Johnson, D. S. (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco.Google Scholar
Goldberg, A. V. and Marchetti-Spaccamela, A. (1984) On finding the exact solution of a zero-one knapsack problem. Proc. 16th Annual ACM Symp. Theory of Computing, 359–368.Google Scholar
Grimmett, G. R. and Mcdiarmid, C. J. H. (1975) On coloring random graphs. Math. Proc. Camb. Phil. Soc. 77, 313–324.Google Scholar
Karmarkar, N. and Karp, R. M. (1982) The differencing method of set partitioning. Report No. UCB/CSD 82/113, Computer Science Division (EECS), University of California, Berkeley.Google Scholar
Karp, R. M. (1972) Reducibility among combinatorial problems. In Complexity of Computer Computations, ed. Miller, R. E. and Thatcher, J. W., Plenum Press, New York, 85–104.Google Scholar
Loulou, R. (1984) Tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling. Math. Operat. Res. 9, 142–150.Google Scholar
Lueker, G. S. (1982) On the average difference between the solutions to linear and integer knapsack problems. In Applied Probability — Computer Science, The Interface, Vol. I, ed. Disney, Ralph L. and Ott, Teunis J., Birkhäuser, Boston, 489–504.Google Scholar
Moser, L. (1970) The second moment method in combinatorial analysis. In Combinatorial Structures and Their Applications, Gordon and Breach, New York.Google Scholar
Shepp, L. A. (1972a) Covering the circle with random arcs. Israel J. Math. 11, 328–345.CrossRefGoogle Scholar
Shepp, L. A. (1972b) Covering the line with random intervals. Z. Wahrscheinlichkeitsth. 23, 163–170.Google Scholar