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European Journal of Operational Research 165 (2005) 416–422

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Positive half-products and schedulingwith controllable processing times

Adam Janiak a, Mikhail Y. Kovalyov b, Wieslaw Kubiak c,*, Frank Werner d

a Institute of Engineering Cybernetics, Wroclaw University of Technology, Wroclaw, Polandb United Institute of Informatics Problems, National Academy of Sciences of Belarus, and Faculty of Economics,

Belarus State University, 220050 Minsk, Belarusc Faculty of Business Administration, Memorial University of Newfoundland, St. John’s, Canada NFA1B 3X5

d Otto-von-Guericke-Universit€at, Magdeburg, Germany

Received 1 November 2002; accepted 1 May 2003

Available online 2 June 2004

Abstract

We study the single machine scheduling problem with controllable job processing times to minimize a linear

combination of the total weighted job completion time and the total weighted processing time compression. We show

that this scheduling problem is a positive half-product minimization problem. Positive half-products make up an

interesting subclass of half-products and are introduced in this paper to provide a conceptual framework for the

problem with controllable job processing times as well as other problems. This framework allows to readily derive in

one fell swoop a number of results for the problem with controllable processing times from more general results ob-

tained earlier for the half-product. We also present fast fully polynomial time approximation schemes for the problem

with controllable processing times. The schemes apply to all positive half-products.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Single machine scheduling; Controllable processing times; Pseudo-Boolean optimization; Fully polynomial time

approximation scheme; Computational complexity

1. Scheduling with controllable processing times

In the problem with controllable processing

times, there are n independent and non-preemptive

* Corresponding author.

E-mail addresses: [email protected] (A. Janiak),

[email protected] (M.Y. Kovalyov), wkubiak@mun.

ca (W. Kubiak), [email protected]

(F. Werner).

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2004.04.012

jobs to be scheduled for processing on a single

machine. All jobs are available for processing at

time zero. The processing time of job j is a variablepj 2 ½0; uj�, j ¼ 1; . . . ; n. A decision maker is to

determine the values of job processing timesp ¼ ðp1; . . . ; pnÞ and a permutation of jobs p so as

to minimize a linear combination TWC ¼Pnj¼1 wjCj þ

Pnj¼1 vjðuj � pjÞ of the total weighted

completion timePn

j¼1 wjCj, where Cj denotes the

completion time of job j, and the total weighted

processing time compressionPn

j¼1 vjðuj � pjÞ.

ed.

mailto:[email protected]

A. Janiak et al. / European Journal of Operational Research 165 (2005) 416–422 417

All numerical data are positive integers. Thevalue of variable pj is a non-negative real number

in ½0; uj�, j ¼ 1; . . . ; n. Setting pj ¼ 0 means that

either the processing time of job j is negligible andthus it practically does not delay the completion

times of other jobs or job j is rejected with penalty

vjuj.Vickson [9,10] was first to study this problem,

as well as a more general problem with arbitrarynon-negative lower bounds lj, lj 6 uj, j ¼ 1; . . . ; n,on job processing times, more than 20 years ago.

Other applications of the problem can be found in

Williams [12] and Janiak [3], where the reader is

referred to for more comprehensive references.

For arbitrary lj and wj ¼ 1, j ¼ 1; . . . ; n, Vick-son [9] recasts the problem as an assignment

problem. For arbitrary weights wj, Vickson [10]presents an enumerative algorithm for the prob-

lem. Vickson [9] also shows that the search for

optimal job processing times p ¼ ðp1; . . . ; pnÞ can

be limited as follows.

Lemma 1. There exists an optimal p ¼ ðp1; . . . ; pnÞwith pj 2 flj; ujg, j ¼ 1; . . . ; n.

Furthermore, the shortest weighted processing

time (SWPT) rule of Smith [8] limits the search for

an optimal permutation p, given p, as follows.

Lemma 2. There exists an optimal p with ppðjÞ=wpðjÞ 6 ppðjþ1Þ=wpðjþ1Þ for j ¼ 1; . . . ; n� 1.

From Lemmas 1 and 2 the following corollaryfollows immediately.

Corollary 1. There exists an optimal solution suchthat pj 2 flj; ujg, j ¼ 1; . . . ; n, jobs with processingtimes pj ¼ lj are sequenced in the non-decreasingorder of lj=wj and jobs with processing times pj ¼ ujare sequenced in the non-decreasing order of uj=wj.

From now on, we assume that the jobs are re-

indexed such that u1=w1 6 � � � 6 un=wn, and lj ¼ 0,

j ¼ 1; . . . ; n. By Corollary 1, the scheduling prob-

lem with controllable job processing times, we also

refer to it as the problem of minimizing TWC for

convenience, reduces to deciding on a partition of

the set of jobs into a subset with pj ¼ 0 and a

subset with pj ¼ uj, and then scheduling the latterjobs in the increasing order of their indices. Let

p� ¼ ðp�1; . . . ; p�nÞ denote an optimal selection of

processing times in the problem of minimizing

TWC.

Our goal is threefold. First, to show in Section

2, that the scheduling with controllable processing

times is polynomially equivalent to the problem of

minimizing a special subclass of half-products. Wedefine the latter in Section 2, it suffices to mention

here that the equivalence immediately implies that

the problem of minimizing TWC is NP-hard in the

ordinary sense. Second, to show in Section 3, the

two fast fully polynomial time approximation

schemes (FPTAS) for the problem of minimizing

TWC (see Garey and Johnson [2]) for the defini-

tion of FPTAS. The schemes generalize a wellknown FPTAS proposed for half-product mini-

mization by Badics and Boros [1]. One runs

in Oðn2 logU=eÞ time, where U ¼P

16 j6 n uj,the other in Oðn2 logW =eÞ time, where W ¼P

16 j6 n wj. Finally, to briefly discuss, in Section 4,

prospects of using the FPTAS developed in this

paper to improve efficiency of the existing

FPTAS’s for a special subclass of positive half-

products.

2. Scheduling with controllable processing times and

half-product minimization

The half-product is a pseudo-Boolean function

of the form

HðxÞ ¼ Hðx1; . . . ; xnÞ

¼ DþX

16 i<j6 n

aibjxixj �X

16 i6 n

cixi;

where xj 2 f0; 1g, j ¼ 1; . . . ; n, a ¼ ða1; . . . ; an�1Þand b ¼ ðb2; . . . ; bnÞ are vectors of non-negative

integers, c ¼ ðc1; . . . ; cnÞ is an arbitrary integervector, and D is an integer. Denote by

x� ¼ ðx�1; . . . ; x�nÞ a 0–1 vector minimizing HðxÞ.The half-product was introduced by Badics and

Boros [1] for D ¼ 0, and independently by Kubiak

[6]. It has attracted attention since a number of

scheduling problems can be recast as half-product

minimization problems (see Kubiak [7]).

418 A. Janiak et al. / European Journal of Operational Research 165 (2005) 416–422

Theorem 1. The problem of minimizing TWC andthe problem of minimizing half-products witha2=b2 6 � � � 6 an�1=bn�1, are polynomially equiva-lent.

Proof. Let vectors w ¼ ðw1; . . . ;wnÞ, u ¼ ðu1; . . . ;unÞ, and v ¼ ðv1; . . . ; vnÞ make up an instance of

the problem of minimizing TWC. Define a half-product as follows:

TWCðxÞ ¼Xn

j¼1

wjxjXj

i¼1

uixi þXn

j¼1

vjujð1� xjÞ

¼X

16 i<j6 n

uiwjxixj �Xn

j¼1

ujðvj � wjÞxj

þXn

j¼1

vjuj; ð1Þ

where obviously u2=w2 6 � � � 6 un�1=wn�1. Let usset xj to 1 if pj ¼ uj and to 0 if pj ¼ 0. By Corollary

1, there always is an optimal p� which translates

this assignment into an optimal x�. Moreover,

both problems have the same optimal value.

Now, let

HðxÞ ¼ DþX

16 i<j6 n

aibjxixj �X

16 i6 n

cixi

be a half-product. Define an instance of the TWC

minimization problem as follows: uj ¼ aj and

wj ¼ Mbj for j ¼ 1; . . . ; n, where M ¼Qn

j¼1 aj,b1 ¼ da1b2a2

e and an ¼ dan�1bnbn�1

e, and vj ¼ Mðbj þ cjajÞ for

j ¼ 1; . . . ; n. The multiplier M is chosen such that

all vj are integer. By definition of b1 and an as wellas inequalities a2=b2 6 � � � 6 an�1=bn�1, we have

that vectors w ¼ ðw1; . . . ;wnÞ, u ¼ ðu1; . . . ; unÞ, andv ¼ ðv1; . . . ; vnÞ make up an instance of the prob-

lem of minimizing TWC with u1=w1 6 � � � 6 un=wn.

Let us set pj to uj if xj ¼ 1 and to 0 if xj ¼ 0. By

Corollary 1, there always is an optimal x� which

translates this assignment into an optimal p�.Moreover, the optimal value of the TWC mini-mization problem is equal to M ½Hðx�Þ � DþP

16 i6 n ðcj þ ajbjÞ�. �

It follows immediately from Theorem 1 that the

TWC minimization is NP-hard in the ordinary

sense since the half-product minimization with

a2=b2 6 � � � 6 an�1=bn�1, is NP-hard (see Jurisch

et al. [4]). Recently, Wan et al. [11] have indepen-dently proved that the problem of minimizing

TWC is NP-hard.

The half-product TWCðxÞ given by (1) admits

a pair of dynamic programming algorithms (see

Jurisch et al. [4]). One runs in OðnPn

j¼1 wjÞ time,

and thus solves the TWC minimization problem

with weights wj ¼ 1, j ¼ 1; . . . ; n, in Oðn2Þ time

which is faster than the assignment algorithm,running in Oðn3Þ time, of Vickson [9]. The latter,

however, solves a more general problem with

arbitrary lj, j ¼ 1; . . . ; n. The other algorithm runs

in OðnPn

j¼1 ujÞ time, and thus solves the problem

with processing times in ½0; 1�, i.e., uj ¼ 1, j ¼1; . . . ; n, in Oðn2Þ time. Finally, it is clear from

the TWCðxÞ definition that vj 6wj, j ¼ 1; . . . ; n,implies p�j ¼ 0, j ¼ 1; . . . ; n.

Badics and Boros [1] derived a FPTAS for the

half-product minimization problem with D ¼ 0.

However, their scheme cannot be directly used as a

FPTAS for the TWC minimization because adding

constant D ¼Pn

j¼1 vjuj can significantly decrease

the absolute value of the optimum for some in-

stances of the half-product minimization problem.

To explain this, we begin with the following result.

Lemma 3. For any positive rational function f of n,there always is an instance of the TWCðxÞ minimiza-tion problem such that jTWCðx�Þ � Dj=jTWCðx�Þj >f ðnÞ, where D ¼

Pnj¼1 vjuj, for an optimal x�.

Proof. We first observe that TWCðx�Þ > 0 and

TWCðx�Þ � D6 TWCð0; . . . ; 0Þ � D ¼ 0. Therefore,inequality in the statement of the lemma can be

written as

D > ðf ðnÞ þ 1ÞTWCðx�Þ:Obviously,X

16 i6 j6 n

uiwj ¼ TWCð1; . . . ; 1ÞPTWCðx�Þ:

Now, consider an instance with vj ¼ 2df ðnÞþ1eðwj þ � � � þ wnÞ, for j ¼ 1; . . . ; n. We have

D ¼ 2df ðnÞ þ 1eTWCð1; . . . ; 1Þ;for this instance and thus the lemma holds. �

Let x0 be an e-approximate solution to the

problem of minimizing TWCðxÞ � D. We have


D ¼ TWCðx0Þ � TWCðx�ÞjTWCðx�Þ � Dj 6 e:

It follows from Lemma 3 that

TWCðx0Þ � TWCðx�ÞTWCðx�Þ > f ðnÞD;

for some instances. Therefore, an e-approximate

solution to the problem of minimizing TWCðxÞ�D obtained by the FPTAS of Badics and Boros [1]

cannot be used to obtain an f ðnÞe-approximatesolution to the problem of minimizing TWCðxÞ forany rational function f ðnÞ, a polynomial in par-

ticular. Consequently, we need a different FPTAS

than that of Badics and Boros. Such a FPTAS is

presented in the following section.

3. Positive half-products and their FPTAS

Consider any half-product

HðxÞ ¼ DþX

16 i<j6 n

aibjxixj �X

16 i6 n

cixi:

Let N ¼ fi : ci < 0g and P ¼ fi : ci P 0g. We

can rewrite HðxÞ as follows:

HðxÞ ¼ D�X

i2Pci þ

X

16 i<j6 n

aibjxixj

þX

i2Pcið1� xiÞ þ

X

i2Nð�ciÞxi;

with all coefficients standing at variables or theirproducts being non-negative. We refer to a half-

product as a positive half-product if the constant

D�P

i2P ci P 0. Thus, the positive half-products

are pseudo-Boolean functions of the form

F ðxÞ ¼X

16 i<j6 n

aibjxixj þXn

j¼1

hjð1� xjÞ

þXn

j¼1

gjxj þ d;

where all coefficients are non-negative integers.

The TWCðxÞ is a positive half-product since wehave TWCðxÞ ¼ F ðxÞ by setting d ¼ 0, aj ¼ uj, bj ¼wj, hj ¼ ujvj and gj ¼ ujwj, j ¼ 1; . . . ; n (see (1)).

We now develop a FPTAS for the problem of

F ðxÞ minimization, which obviously directly ap-

plies to the problem of minimizing TWC. We startwith a simple decomposition result for F ðxÞ (see

also Badics and Boros [1]).

Lemma 4. For any x and k ¼ 1; . . . ; n, we have:

F ðxÞ ¼ F1;kðxÞ þ a1;kðxÞbkþ1;nðxÞ þ Fkþ1;nðxÞ þ d;

where

F1;kðxÞ ¼X

16 i<j6 k

aibjxixj þXk

j¼1

hjð1� xjÞ þXk

j¼1

gjxj;

Fkþ1;nðxÞ ¼X

kþ16 i<j6 n

aibjxixj þXn

j¼kþ1

hjð1� xjÞ

þXn

j¼kþ1

gjxj;

a1;kðxÞ ¼Xk

j¼1

ajxj;

bkþ1;nðxÞ ¼Xn

j¼kþ1

bjxj:

Proof. Straightforward algebraic manipula-

tion. �

Though F ðxÞ is a pseudo-Boolean function on

binary vectorswe rather see it as a function on finite

words over two letter alphabet f0; 1g in our sub-

sequent presentation, which needs to discuss binary

vectors of varying dimension. Let f0; 1g� be the setof all finite words on the alphabet {0, 1} with the

empty word K included. Let jxj be the length of x,i.e., the number of letters in x 2 f0; 1g�. For a wordx ¼ x1x2 . . . xn, let us call the word x1x2 . . . xk, thek-prefix of x, and the word xkþ1 . . . xn, the ðn� kÞ-suffix of x, for k ¼ 0; 1; . . . ; n. We assume 0-prefix

and 0-suffix being empty words. The concatena-

tions x0 and x1 denote word x extended by 0 and 1,respectively.

Our FPTAS trims the solution space using a

general approach developed by Badics and Boros

[1], and Kovalyov and Kubiak [5] for half-prod-

ucts and decomposable partition problems. The

scheme takes an instance of a positive half-product

and e as its inputs, and iteratively, starting with the


empty word K, builds a solution to the half-product minimization. At iteration k selected

words of length k are partitioned into subsets to

ensure that each subset includes only those words

that are d-close to each other, more precisely, for

any two words x and y in the same subset the

algorithm ensures

ja1;kðxÞ � a1;kðyÞj6 dminfa1;kðxÞ; a1;kðyÞg

for some positive d dependent on e and n to bedefined later. Then, F1;kð�Þ is used to select a single

word x from each subset of the partition. The word

has the smallest value F1;kð�Þ among all words in

the same subset of the partition. Only the selected

words pass to iteration k þ 1, where each word is

extended by concatenating either 0 or 1 at its end,

and the iteration repeats. Finally, when k reaches nthe algorithm stops selecting a word with theminimum value of F ð�Þ among all words that

reached iteration n. The details of the algorithm

are as follows:

� a1;kðx ÞÞbkþ1;nðx Þ:

Algorithm Ae

Step 1 (Initialization). Calculate d > 0 such that

ð1þ dÞn ¼ 1þ e. Set k ¼ 0 and X0 ¼ fKg.Step 2 (Recursive filtering). Construct set Yk ¼

fx0; x1jx 2 Xk�1g. Calculate a1;kðxÞ and

F1;kðxÞ for each x 2 Yk. If k ¼ n, then set

Xn ¼ Yn and go to Step 3. Otherwise, parti-

tion Yk into subsets Yr;k, r ¼ 1; . . . ; sk, suchthat

ja1;kðxÞ � a1;kðyÞj6 d minfa1;kðxÞ; a1;kðyÞg;for any x and y in the same subset. From

each subset Yr;k, select a vector xr;k such

that F1;kðxr;kÞ ¼ minfF1;kðxÞjx 2 Yr;kg. Set

Xk ¼ fxr;kjr ¼ 1; . . . ; skg, k ¼ k þ 1 and

repeat Step 2.

Step 3 (e-approximate solution). Select a solutionxe 2 Xn such that F ðxeÞ ¼ minfF ðxÞjx 2Xng and stop. �

We now show that algorithm Ae produces

solution xe of required relative error e. The algo-

rithm complexity is shown in Theorem 3, where

the efficient implementation of Step 2 is discussed

in detail.

Theorem 2. Algorithm Ae finds xe 2 Xn such thatF ðxeÞ � F ðx�Þ6 eF ðx�Þ.

Proof. For an optimal x�, let xð0Þ; . . . ; xðnÞ be nþ 1

words of length n each such that

(a) k-prefix of xðkÞ is in Xk, for k ¼ 0; . . . ; n,(b) both xðkÞ and x� share the same ðn� kÞ-suffix,

for k ¼ 0; . . . ; n,(c) k-prefixes of xðk�1Þ and xðkÞ are in the same Yr;k,

for k ¼ 1; . . . ; n.

By (a) and (b), xð0Þ ¼ x�.Our proof relies on inequalities (2), (3) and (5)

that we now prove.

First, since all coefficients in F ðxÞ are non-neg-

ative, we have

a1;kðx�Þbkþ1;nðx�Þ6 F ðx�Þ: ð2ÞSecond, we have

a1;kðxðk�1ÞÞ6 ð1þ dÞk�1a1;kðx�Þ; k ¼ 1; . . . ; n: ð3ÞWe prove this inequality by induction on k. Fork ¼ 1, (3) holds since xð0Þ ¼ x�. Assume that (3)

holds for 16 k6 n� 1. Let us prove that (3) holdsfor k þ 1. By (c), k-prefixes of xðk�1Þ and xðkÞ are inthe same subset Yr;k, thus we have

a1;kðxðkÞÞ6 ð1þ dÞa1;kðxðk�1ÞÞ; k ¼ 1; . . . ; n: ð4ÞFinally,

a1;kþ1ðxðkÞÞ ¼ a1;kðxðkÞÞ þ akþ1x�kþ1

6 ð1þ dÞa1;kðxðk�1ÞÞ þ akþ1x�kþ1

6 ð1þ dÞka1;kðx�Þ þ akþ1x�kþ1

6 ð1þ dÞka1;kþ1ðx�Þ:Here, the first equation follows from the defini-tions of Lemma 4, the first inequality follows from

(4), the second one follows from the inductive

assumption, and the last one again from the defi-

nitions of Lemma 4.

Third, we have

F ðxðkÞÞ � F ðxðk�1ÞÞ6 dð1þ dÞk�1F ðx�Þ: ð5ÞTo prove it, we observe that by definitions of F ðxÞand xðkÞ, we have

F ðxðkÞÞ� F ðxðk�1ÞÞ ¼ F1;kðxðkÞÞ� F1;kðxðk�1ÞÞþ ða1;kðxðkÞÞðk�1Þ �


By (c), k-prefixes of xðk�1Þ and xðkÞ are in the same

subset Yr;k. Consequently, ða1;kðxðkÞÞ � a1;kðxðk�1ÞÞÞ6 da1;kðxðk�1ÞÞ. Moreover, the minimum value of

F1;k over all vectors in Yr;k is attained at xðkÞ, thus,F1;kðxðkÞÞ6 F1;kðxðk�1ÞÞ. Therefore, by (2) and (3),

F ðxðkÞÞ � F ðxðk�1ÞÞ6 da1;kðxðk�1ÞÞbkþ1;nðx�Þ6 dð1þ dÞk�1a1;kðx�Þbkþ1;nðx�Þ6 dð1þ dÞk�1F ðx�Þ:

We are now ready to prove the theorem. We have

F ðxeÞ6 F ðxðnÞÞ and xð0Þ ¼ x�. Therefore, by (5) and

the definition of d, ð1þ dÞn ¼ 1þ e, we have

F ðxeÞ � F ðx�Þ6 F ðxðnÞÞ � F ðxð0ÞÞ

¼Xn

k¼1

ðF ðxðkÞÞ � F ðxðk�1ÞÞÞ

6 dF ðx�ÞXn

k¼1

ð1þ dÞk�1 ¼ eF ðx�Þ;

which completes the proof. �

Theorem 3. Algorithm Ae can be implemented torun in Oðn2 logA=eÞ time, where A ¼

Pnj¼1 aj.

Proof. The key to the complexity of Ae is the

implementation of set Yk partitioning in Step 2.

There, we arrange the words in Yk in ascending

order of their a1;kð�Þ values, we call this order ana-order, so that 06 a1;kðy1Þ6 a1;kðy2Þ6 � � � 6a1;kðyjYk jÞ. Then, we assign y1; y2; . . . ; yi1 to set Y1;kuntil detecting i1 such that a1;kðyi1Þ6 ð1þ dÞa1;kðy1Þand a1;kðyi1þ1Þ > ð1þ dÞa1;kðy1Þ. If such an i1 does

not exist, then we set Y1;k ¼ Yk and stop. Next, we

assign yi1þ1; yi1þ2; . . . ; yi2 to set Y2;k until detecting i2such that a1;kðyi2Þ6 ð1þ dÞa1;kðyi1þ1Þ and a1;kðyi2þ1Þ> ð1þ dÞa1;kðyi1þ1Þ. If such an i2 does not exist,then we set Y2;k ¼ Yk n Y1;k and stop. We continue

this partitioning until yjYk j is included in Ysk ;k, forsome sk. It is crucial for the complexity to notice

here that, if Xk�1 is a-ordered, then obviously both

fx0jx 2 Xk�1g and fx1jx 2 Xk�1g easily inherit its a-order and their merging leads to the a-ordered set

Yk in linear time. Moreover, the selection of a

single vector from each set of the partitionY1;k; . . . ; Ysk ;k again inherits the a-order which

results in a-ordered Xk. Consequently, the a-order

of words is an invariant of Step 2, and therefore the

step can be implemented in OðjYkjÞ, and the whole

algorithm in OðPn

k¼1 jYkjÞ time. Furthermore, we

have jYkj ¼ 2jXk�1j ¼ 2sk�1and sk 6K þ 1, k ¼ 1;. . . ; n, where K is an integer that satisfies

ð1þ dÞK PA. Consequently, the algorithm runs in

OðnKÞ time, and it remains to estimate the value of

K. We have KP logA= logð1þ dÞ. From therelationship between e and d defined in Step 1 of

the algorithm, we have logð1þ dÞ ¼ logð1þ eÞ=n.Since logð1þ eÞ6 e, for 0 < e6 1, then KPn logA=e. Notice that if e > 1, then a 1-approxi-

mate solution can be taken as an e-approximate

solution, and we may assume 0 < e6 1 without

loss of generality. Thus, by setting K ¼ dn logA=ee,we obtain the required complexity. �

Theorems 2 and 3 prove that Ae is a FPTAS for

any positive half-product, and the problem with

controllable job processing times in particular.

Another scheme with time complexity

Oðn2 logB=eÞ, where B ¼Pn

j¼1 bj, can be derived in

a similar way as Ae. The scheme relies on values

bkþ1;nðxÞ and Fkþ1;nðxÞ for its recursive filtering inStep 2, and builds word xnxn�1 . . . x1 starting from

empty word K.

4. Conclusions and further research

We have shown that the single machine sched-

uling problem with controllable job processingtimes is polynomially equivalent to the problem of

maximizing a special subclass of half-products,

namely, positive half-products. This immediately

proves that not only is the former problem NP-

hard but also that it can be solved in pseudo-

polynomial time by dynamic programs proposed

earlier for the half-product minimization, see [4].

We have also developed a couple of fully polyno-mial time approximation schemes for the problem

with controllable processing times. The schemes

apply to a general class of problems called positive

half-products that we have also introduced in this

paper. The class includes, for instance, the two

machine weighted completion time problem, and it

is very likely to include many more scheduling

problems. The search for them seems an exciting


and practically important topic for further re-search since it may ultimately lead to more efficient

approximation schemes, based on the schemes

presented in this paper, for many scheduling

positive half-product problems.

Acknowledgements

M.Y. Kovalyov was supported in part by

INTAS under grant number 00-217. W. Kubiak

has been supported by the Natural Sciences and

Engineering Council of Canada Research GrantOGP0105675. The authors would like to thank

anonymous referees for their constructive com-

ments that resulted in an improved paper.

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