On the rate of convergence to optimality of the LPT rule

On the rate of convergence to optimality of the LPT rule

Citation for published version (APA):

Frenk, J. B. G., & Rinnooy Kan, A. H. G. (1985). On the rate of convergence to optimality of the LPT rule. (Memorandum COSOR; Vol. 8501). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985

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januari 1985

Memorandum CaSaR 85-01

On the rate of convergence to optimality of the LPT rule

by

J.B.G. Frenk A.H.G. Rinnooy Kan

1

ON THE RATE OF CONVERGENCE TO OPTIMALITY OF THE LPT RULE

*

J.B.G. Frenk

(Department of Industrial Engineering and Operations Research, University of California, Berkeley)

A.R.G. Rinnooy Kan

(Econometric Institute, Erasmus University, Rotterdam)

Abstract

The LPT rule is a heuristic method to distribute jobs among identical machines so as to minimize the makes pan 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 show that the speed of convergence is proportional to logn/n, thus

extending earlier results obtained for the uniform and exponential distribution.

* Present address first author: Eindhoven University of Technology,

Department of Mathematics and Computing Science, Eindhoven.

Suppose that n jobs with processing times PI'." 'Pn have to be distributed among m identical machines. If the sum of the processing times assigned to machine i is denoted by Z (i) (i-1, ••• ,m), then a common objective is to minimize the makespan

z(m~

=

maxi{Z (i)}. For this NP-hard problem many

n n

heuristics have been proposed and analyzed; we refer to [Graham et ale 1979] for a survey. Among them, the LPT rule in which jobs are assigned to the first available machine in order of decreasing Pj is a particularly simple and

attractive one. The value Z(m)(LPT) produced by this rule is related to the

n

optimal solution value Z(m)(OPT) by [Graham 1969]

n Z(m) (LPT) n

<.!_..L.

z(m)(OPT) - 3 3m n (1)

Computational evidence, however, suggests that this worst case analysis is unnecessarily pessimistic in that problem instances for which (1) is satisfied as an equality appear to occur only rarely.

To achieve a better understanding of this phenomenon, let us assume the processing times Pj(j=l, ••• ,n) to be independent, identically distributed random variables. The relation between the random variables z(m)(OPT) and

-n

~m)(LPT)

can then be subjected to a probabilistic analysis. In [Frenk &

Rinnooy Kan 1984J it was shown that (under mild conditions on the distribution of the Pj) the absolute error

(2)

converges to 0 almost surely as well as in expectation. Thus, the heuristic is aSymptotically optimal in a strong (absolute rather than relative) sense, which provides an explanation for its excellent computational behaviour.

In [Frenk & Rinnooy Kan 1984], the speed at which the absolute error converges to 0 was analyzed for the special cases of the uniform and exponential

distribution respectively. Here we extend these results by showing that for a large class of distributions this speed is proportional to log nino This implies that, although the absolute optimality of the LPT rule could only be established asymptotically, the convergence of the absolute error to 0 at

3

least occurs reasonably fast.

The main result is described and proved in Section 2. Some extensions and conjectures are briefly examined in Section 3.

In this section we shall assume that the distribution function F of the processing times satisfies F(O) = 0 and moreover has a strictly positive continuous density function f on [0,£] for some £

>

O. We also assume the first moment E.£.. to be finite.

Theorem If F satisfies the above conditions, then

lim sup n (Z(m)(LPT) - Z(m)(OPT» < ~.

n+oo log n -n -n (3)

Proof Let

~(n)

>

~(n-l)

> ••• >

~(1)

be the order statistics of the processing times. As demonstrated in [Frenk & Rinnooy Kan 1984], the absolute error of the LPT rule is bounded (up to a constant) by

(4)

for some constant a. Our analysis will focus on the asymptotic behaviour of this random variable.

We first observe that

(k) 'i'k (j)

max1<k<[En/2){ap - Lj=I P }

+

+

max {ap(n)_ \~En/21p(j) O}

LJ=l ' (5)

In [Frenk

&

Rinnooy Kan 1984], we showed that for every E

>

0

(a.s.) (6)

which implies that, if den) = n/log n, then

(a.s.) (7)

5

Since <..E.(n), ••• ,..E,.(l)

g

(F+(U(n», ••• ,F+(U(l», where F+(y) =

sup {xIF(x)

i

y} and ..!!.(n), ••• ,U(l) are the order statistics of n independent random variables uniformly distributed on [0,1], we have that

{ (k) ,k (j) }

Pr max₁(k([ e:n/2] {ap - Lj=l.E. }

2

den)

i

i

pr{max

1(k([e:n/2] {aF+(u(k» -

I~=1

F+(U(j»}

2

den)

/I.

.,g( [

e:n/ 2])

.5.

e:}

+

From [Albers et ale 1976], we know that

and hence

Hence, we now focus on the first term on the right hand side of (8).

(8)

(9)

(10)

Our assumptions on F imply that there exist positive constants a and A such that, for e: sufficiently small,

+

ay

.5.

F (y)

.5.

Ay (11 )

for all y € [O,e:]. Hence, for all k € {1, •••• [e:n/2]},

(12)

if U( [e:n/ 2]) ( _{e:, so that}

{ {aF+( (k) _ ,k. F+(U(j»}

>

den)

prmax1(k([e:n/2]

.,g

L

-A U([En/2]) < E} <

-

-( 13)

with a* = aA/a and d*(n) = d(n)/a.

Thus we are now back in the uniform case. As in [Frenk & Rinnooy Kan 1984], we rewrite the right hand side of (13) as

(14)

with

~j =

Ii=1

Et'

and

Et

exponentially distributed with parameter 1. We now condition on~1 being greater or smaller than (n+l)/2. Since, from a

generalization of Chebyshev's inequality, for all A > 0

= (

exp(A/2»)n+1 <

1+1.

-<

(1 + A exp( A/2)/2 )11+1

- 1

+

A '

we may choose A so that exp(A/2) < 2 to conclude that

pr{~1 < (n+l)/2} < yn+l with A < 1 and concentrate on

pr{max₁(k<[En/2] {a*1k -

I~=l ~j} ~

d*(n)(n+l)/2}

~

< I[En/2J pr {a*q -

I~

q.

>

d*(n)(n+l)/2}

=

- k=1 -% J=1 J

-= ,[En/2]pr{,k (a*+t-k-l)r

> d*(n)(n+1)/2}

Lk=l Lt=l R. -(15) (16) (17)

Analogously to (15), the probabilities occurring in (17) with k < [a*] + 1 can be bounded by

7

C1(a*) exp(-Ad*(n)(n+l)/2) (18)

and the probabilities with k

>

[a*] + 1 by -k

C

2(a*,A)(1+A) exp(-Ad*(n)(n+l)/2) (19)

where C1(a*) and C2(a*,A) are constants only depending on a* and A. Hence, their sum can be bounded by

C

3(a*,A) exp(-Ad*(n)(n+l)/2) (20)

and we conclude that, for some constant C,

The result in the preceding section establishes that under conditions slightly more stringent than those under which the absolute LPT error converges to 0

(a.s.), the speed of convergence corresponds to what was established for the uniform and exponential case only in [Frenk & Rinnooy Kan 1984]. The result confirms one's intuition that the speed of convergence is about l/n whenever the small order statistics of the processing times behave as they do in the uniform case.

Several extensions seem possible. E.g., we conjecture that the result will also hold if we only require that f(O)

=

0, f(x)

>

0 on (O,e], provided that f(x) converges to 0 appropriately when x ~ 0 (say, f(x)

=

O(x

B

».

In fact, we think that under quite general conditions the speed of convergence can be shown to be l/n rather than logn/n, as we indicated already above. Finally, we are confident that these results also hold if we consider convergence in

expectation rather than almost surely, and that they extend easily to the case of uniform rather than indentical machines.

Our results, as well as other recent ones in this area (e.g. [Boxma 1984]), confirm the remarkable amenability of the LPT rule to probabilistic analysi~

of this type. Extensions to more complicated heuristics are worthy of pursual.

REFERENCES

W. Albers, P.J. Bickel & W.R. van Zwet (1976), 'Asymptotic expansions for the power of distribution free tests in the one-sample problem', Annals of Statistics!!. (1), p. 108-156.

J.B.G. Frenk, A.H.G. Rinnooy Ran (1984), 'The asymptotic optimality of the LPT rule', Technical Report Econometric Institute, Erasmus University

Rotterdam.

R.L. Graham (1969), 'Bounds on multiprocessing timing anomalies', SIAM J. Appl. Math.JZ., 263-269.

R.L. Graham, E.L. Lawler, J.K. Lenstra & A.H.G. Rinnooy Kan (1979), 'Optimization and approximation in deterministic sequencing and scheduling: a survey', Ann. Discrete Math.

2.,

287-326.