The rate of convergence to optimality of the LPT rule

The LPT rule is a heuristic method to distribute jobs among identical machines so as to minimize the makespan of the resulting schedule. If the processing times of the jobs are assumed to be independent identically distributed random variables, then (under a mild condition on the distribution) the absolute error of this heuristic is known to converge to 0 almost surely. In this note we analyse the asymptotic behaviour of the absolute error and its first and higher moments to show that under quite general assumptions the speed of convergence is proportional to appropriate powers of (log log n)/n and 1/n. Thus, we simplify, strengthen and extend earlier results obtained for the uniform and exponential distribution.

On the rate of convergence to optimality of the LPT rule - postscript

This postscript contains the proofs of two results listed in the paper 'On the rate of convergence to optimality of the LPT rule'

The asymptotic behaviour of a distributive sorting method

In the distributive sorting method of Dobosiewicz, both the interval between the minimum and the median of the numbers to be sorted and the interval between the median and the maximum are partitioned inton/2 subintervals of equal length; the procedure is then applied recursively on each subinterval containing more than three numbers. We refine and extend previous analyses of this method, e.g., by establishing its asymptotic linear behaviour under various probabilistic assumptions.sorting;probabilistic analysis

Probabilistic analysis of algorithms for dual bin packing problems

In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and reveal the connection between this problem and the classical bin packing problem as well as to renewal theory.

Order statistics and the linear assignment problem

Under mild conditions on the distribution functionF, we analyze the asymptotic behavior in expectation of the smallest order statistic, both for the case thatF is defined on (–, +) and for the case thatF is defined on (0, ). These results yield asymptotic estimates of the expected optiml value of the linear assignment problem under the assumption that the cost coefficients are independent random variables with distribution functionF.asymptotic analysis;linear assignment problem;order statistic