Convex Programming for Scheduling Unrelated Parallel Machines

Transcription

1 Convex Prograing for Scheduling Unrelated Parallel Machines Yossi Azar Air Epstein Abstract We consider the classical proble of scheduling parallel unrelated achines. Each job is to be processed by exactly one achine. Processing job j on achine i requires tie p ij. The goal is to find a schedule that iniizes the l p nor. Previous work showed a 2-approxiation algorith for the proble with respect to the l nor. For any fixed l p nor the previously known approxiation algorith has a perforance of θ(p). We provide a 2-approxiation algorith for any fixed l p nor (p > 1). This algorith uses convex prograing relaxation. We also give a 2-approxiation algorith for the l 2 nor. This algorith relies on convex quadratic prograing relaxation. To the best of our knowledge, this is the first tie that general convex prograing techniques (apart fro SDPs and CQPs) are used in the area of scheduling. We show for any given l p nor a PTAS for any fixed nuber of achines. We also consider the ultidiensional generalization of the proble in which the jobs are d-diensional. Here the goal is to iniize the l p nor of the generalized load vector, which is a atrix where the rows represent the achines and the coluns represent the jobs diension. For this proble we give a (d + 1)-approxiation algorith for any fixed l p nor (p > 1). 1 Introduction We consider the classical proble of scheduling jobs on parallel unrelated achines. Lenstra et. al [14] and Shoys and Tardos [16] provided a 2-approxiation algorith for iniizing the akespan (l nor). However, for the l p nor only θ(p)-approxiation algorith was known (see [2]). We provide a 2-approxiation algorith for any l p nor. In addition we show a 2-approxiation algorith for the l 2 nor. Our approxiation algoriths are based on convex prograing relaxations. To the best of our knowledge, this is the first tie that general convex prograing techniques (apart fro SDPs and CQPs) are used in the area of scheduling. Seidefinite prograing (SDP) and convex quadratic prograing (CQP) are special cases of convex prograing (CP). Given any ǫ > 0 convex progras can be solved within an additive error of ǫ under soe requireents on the convex objective function and on the feasible space. This can be done through the ellipsoid algorith (Grötschel et. al [9]) and ore efficiently using interior-point ethods (see e.g, Nesterov and Neirovsky [15]). Linear prograing (LP) relaxations have been proved to be a useful tool in the design and analysis of approxiation algoriths for any graph and cobinatorial probles. There are several applications of linear prograing relaxations for achine scheduling probles, see, e.g., [14, 16, 10, 4, 5]. The iportance of seidefinite prograing is that soeties it leads to tighter relaxations for any graph and cobinatorial probles. Grötchel et. al [8] used seidefinite prograing to design a polynoial tie algorith for finding the largest stable set in a perfect graph. Goeans and School of Coputer Science, Tel-Aviv University. azar@tau.ac.il. Supported in part by Geran-Israeli Foundation and by the Israel Science Foundation. School of Coputer Science, Tel Aviv University. airep@tau.ac.il. Supported in part by the Geran-Israeli Foundation and by the Israel Science Foundation. 1

2 Williason [7] were the first to use seidefinite prograing relaxations in the design of approxiation algoriths. They used seidefinite relaxations for the probles MAXCUT, MAXDICUT and MAX2SAT. Convex quadratic prograing and seidefinite prograing relaxations in the area of scheduling were first used by Skutella [17]. He used convex quadratic prograing relaxations to design approxiation algoriths for the proble of scheduling unrelated parallel achines so as to iniize total weighted copletion tie of jobs. Techniques: In the design of the 2-approxiation algorith for scheduling jobs on unrelated achines we forulate the proble as a convex progra, where the objective function is the l p nor (p > 1). The obvious CP relaxation yields a large integrality gap. To overcoe this proble we odify the objective function of the CP. The new objective function yields a bounded gap. Our rounding technique is based on the rounding technique of Shoys and Tardos for the generalized assignent proble [16]. They used this rounding technique for the case of l nor. However, they did not need to use the extra property of not increasing the cost. Interestingly in our rounding ethod for the l p nor we need to use this extra property. We use their technique in an enhanced anner using the additional ter in the odified objective function of the convex progra as the cost of the fractional weighted atching of the bipartite graph they for. Then we exploit the property That there exists an integral atching with no greater cost. This atching defines the schedule of the jobs. For the l 2 nor we use the sae convex progra together with randoized rounding to obtain a 2-approxiation algorith. For any l p nor we also design a slightly better than a 2-approxiation algorith for the restricted assignent odel by using a bootstrapping of a 2-approxiation algorith. Additionally we use convex prograing relaxation in the design of a PTAS to the proble. 1.1 Proble Definition We have parallel achines and n independent jobs, where job j takes positive integral processing tie p ij when processed by achine i. The load of achine is the total processing tie required for the jobs assigned to it. The cost of an assignent for an input sequence of jobs is defined as the l p nor of the load vector. Specifically, the l nor is the akespan (or axiu load) and the l 2 nor is the Euclidean nor, which is equivalent to the su of the squares of the load vector. The goal of an assignent algorith is to assign all the jobs so as to iniize the cost. Consider for exaple the case where the weight of a job corresponds to its achine disk access frequency. Each job ay see a delay that is proportional to the load on the achine it is assigned to. Then the average delay over all disk accesses is proportional to the su of squares of the achines loads (naely the l 2 nor of the corresponding achines load vector) whereas the axiu delay is proportional to the axiu load. 1.2 Our Results We show the following results for scheduling jobs on unrelated achines: A 2-approxiation polynoial algorith for the general proble for any given l p nor (p > 1). This iproves the previous θ(p) approxiation algorith given in [2]. A 2-approxiation polynoial algorith for the l 2 nor for the proble, iproving the previous approxiation algorith given in [2]. A PTAS for fixed nuber of achines for any given l p nor with space which is polynoial in both 1 ǫ and (and the input size). We also consider a generalization of the proble and a special case of the proble and obtain the following results: 2

3 We consider the case in which jobs are d-diensional and show a d + 1-approxiation algorith. We consider the restricted assignent odel where each job has a size and should be assigned to one out of soe subset of achines. For this proble we show a slightly better than a 2-approxiation algorith for any l p nor. 1.3 Previous Results Lenstra et. al [14] and Shoys and Tardos [16] presented a 2-approxiation algorith for the akespan, however their algorith does not guarantee any constant approxiation ratio to optial solutions for any other nors (it is easy to coe up with a concrete exaple to support that). For any given l p nor the only previous approxiation algorith for unrelated achines, presented by Awerbuch et al. [2], has a perforance of θ(p) (this algorith was presented as an on-line algorith). Our ain result is a 2-approxiation polynoial algorith for any given l p nor (p > 1). It is also known that there is no approxiation polynoial algorith for the l nor with ratio better then 3/2 (see [14]). In addition the proble is AP X-Hard for any fixed p (see [3]). These hardness results hold for the restricted assignent odel as well. For the l 2 nor the only previous approxiation algorith for unrelated achines, presented by Awerbuch et al. [2], has a perforance of (this algorith was presented as an on-line algorith). We iprove this result by providing a 2-approxiation polynoial algorith for the l 2 nor. Recall that for the l p nor (p > 1) the proble of scheduling in the restricted assignent odel is APX-hard. Thus, there is no PTAS for the proble unless (P = NP ). However, if the nuber of achines is fixed a PTAS can be achieved. We present a PTAS for any given nor and any fixed nuber of achines with better space coplexity then the FPTAS presented for the proble in [3]. Note that for iniizing the akespan Horowitz and Sahni [13] presented a FPTAS for any fixed nuber of achines. Lenstra et al. [14] suggested a PTAS for the sae proble (i.e. iniizing the akespan) with better space coplexity. Our algorith for the restricted assignent odel has an approxiation ratio of 2 Ω( 1 2 p ). Previously, Azar et al. [3] presented an all-nor 2-approxiation polynoial algorith which provides a 2-approxiation to all nors siultaneously. Their algorith gives a 2-approxiation to any fixed l p nor. Our iproved algorith uses their algorith recursively. For d-diensional jobs we present a (d + 1)-approxiation algorith for any fixed diension d and any fixed l p nor (p > 1). This is in contrast to the θ(p) algorith that can be obtained fro [2]. Other related results: Other scheduling odels have also been studied. For the identical achines odel, where each job has an associated weight and can be assigned to any achine, Hochbau and Shoys [12] presented a PTAS for the case of iniizing the akespan. Later, Alon et al. [1] showed a PTAS for any l p nor in the identical achines odel. For the related achines odel, in which each achine has a speed and the achine load equals the su of jobs weights assigned to it divided by its speed, Hochbau and Shoys [11] presented a PTAS for the case of iniizing the akespan. Epstein and Sgall [6] showed a PTAS for any l p nor in the sae odel. 1.4 Paper Structure In Section 3 we present our approxiation algorith for any l p nor. In Section 4 we present an approxiation algorith for the l 2 nor. In section 5 we give for any given l p nor a slightly better approxiation algorith for the restricted assignent odel. In section 6 we construct for any given l p nor a PTAS for any fixed nuber of achines. In Section 7 we consider the ultidiensional 3

4 generalization of the proble and for this proble we present an approxiation algorith for any l p nor. 2 The Approxiation Algorith (p > 1) 2.1 Convex Prograing Forulation We define the following iniization proble in the unrelated achines odel, where there is a fixed processing tie p ij associated with each achine i = 1,..., and each job j = 1,...,n. Integer solutions to the following convex progra (CP1) give the optial schedules. in t p i x ij = 1 for j = 1,...,n x ij p ij t i = 0 for i = 1,..., x ij 0 for j = 1,...,n, i = 1,...,, where for each achine i = 1,..., and each job j = 1,...,n the variable x ij denotes the relative fraction of job j on achine i and for each achine i = 1,..., the variable t i denotes the load of achine i. We denote the optiu integer solution by OPT p, which is the optiu l p nor to the power of p and OPT is the optiu l p nor. Because of solving convex progra requireents, we change the linear equality constraints of the convex progra to linear inequality constraints, which does not change the optial solution value as follows x ij 1 for j = 1,...,n x ij p ij t i 0 for i = 1,..., x ij 0 for j = 1,...,n, i = 1,...,. This CP relaxation has large integrality gap. To overcoe this proble we odify the objective function in a way which also helps us in our rounding technique as follows. Let g(t ) = tp i be the original objective function. We call it the l p nor function. Let c(x ) = n x ijp p ij be the cost function. Our odified objective function is f(x, t ) = g(t ) + c(x ) and the objective is in f(x, t ). We call the odified convex progra (CP2). Solvability of Convex Progra: Convex progra can be solved within an additive error of ǫ in polynoial tie for any given ǫ > 0 when the feasible region is a convex copact set with nonepty interior (i.e. nonzero volue) under soe requireents (see, e.g., [9, 15]). Specifically, it can be solved for iniizing an objective function which is convex and differentiable if we have a (polynoial tie) separation oracle for the feasible set and the possibility to copute the objective and its subgradient at a given point (in polynoial tie). When these requireents hold we can use for exaple the ellipsoid algorith to solve the optiization iniization proble. To ipleent the ellipsoid algorith we use the separation oracle for the feasible set and the fact that the objective 4

5 function is convex and differentiable. Specifically, we can copute the subgradient, at any given point to obtain a separation half-space to exclude points that are not in the optial set. Since all the constraints in (CP2) are linear inequalities the feasible region is convex. Moreover, the separation oracle is easy to ipleent. Also it is easy to see that the feasible region has a positive volue. The objective function is su of convex functions and hence convex. Moreover, it is easy to copute its value and its subgradient at a given point. Hence, (CP2) is solvable in polynoial tie for any accuracy. We note that in the first proof we show how to overcoe the additive error of ǫ that results fro solving the convex progra. In all other proofs we assue that we get the exact solution to the convex progra, since we can easily overcoe the additive error of ǫ as done in the first proof. Lea 2.1 Let x, t be an optiu solution to (CP2), with value f(x, t ). Then f(x, t ) 2OPT p. Proof: A possible feasible solution to (CP2) is the optiu integer solution to (CP1) denoted by x 0, t 0. For this solution g(t 0) = OPTp, c(x 0 ) OPTp and f(x, t ) f(x 0, t 0), where the first inequality follows fro the fact that for a feasible integer solution to (CP1) denoted by x, holds t g(t ) = (t i )p = ( x ij p ij) p x ij pp ij = c(x ). Hence f(x, t ) f(x 0, t 0) = g(t 0) + c(x 0) 2OPTp. This copletes the proof. 2.2 Convex Prograing Rounding Theore 2.1 The fractional solution to (CP2) can be rounded in polynoial tie to an integral assignent which gives a value which is at ost twice of the optiu for the l p nor. Proof: Given the fractional assignent {x ij } we will show how to construct the desired integral assignent {ˆx ij } in polynoial tie. We use the sae rounding algorith used by Shoys and Tardos for the Generalized Assignent Proble [16]. They showed how to convert a fractional solution {x ij } to an integer solution {ˆx ij } which satisfies the following ˆt i t i + q i ˆx ij c ij x ij c ij, where q i is the size the largest job assigned to achine i by the integral assignent {ˆx ij } and c ij is the cost of job j on achine i. We define c ij = p p ij. Then we obtain x ij c ij = x ij p p ij which is the cost ter in the objective function of (CP2). Now we use the sae rounding procedure defined above with the linear cost function defined by the coefficients c ij. Next we show that the l p nor of the schedule constructed is at ost twice of the optiu g(ˆt ) = ˆt p i (t i + q i ) p 2 p 1 (t p i + qp i ) = 2p 1 ( t p i + q p i ) 2 p 1 ( t p i + ˆx ij p p ij ) 2p 1 ( t p i + = 2 p 1 f(x, t ) 2 p 1 (2OPT p + ǫ) = 2 p OPT p + 2 p 1 ǫ. 5 x ij p p ij )

6 The first inequality follows fro the fact that the load on each achine i is not greater then t i + q i. The second inequality follows fro the inequality (x + y) p 2 p 1 (x p + y p ). The fourth inequality follows fro the fact that ˆx ij p p ij x ij p p ij. The last inequality follows fro Lea 2.1 and the fact that we get an optial solution to (CP2) up to an additive error of ǫ. By choosing ǫ = 1 2 p we get g(ˆt ) 2 p OPT p Since g(ˆt ) and OPT p are positive integral nubers we obtain This copletes the proof. (g(ˆt )) 1 p 2OPT. The proof of the following theore appears in the Appendix. Theore 2.2 ) The approxiation ratio provided by the approxiation algorith is lower bounded by 2 O. ( lnp p 3 Approxiation Algorith (p = 2) The odified objective function of (CP2) for the case p = 2 is in t 2 i + x ij p 2 ij, which gives a convex quadratic progra (CQP) that can be solved within an additive error of ǫ in polynoial tie for any given ǫ > 0. We use randoized rounding: each job j is assigned independently at rando to one of the achines with probabilities given through the values {x ij }; notice that x ij = 1. We denote the obtained integral assignent by {ˆx ij }. Lea 3.1 For the solution {ˆx ij } obtained by randoized rounding: E[ˆt 2 i ] (E[ˆt i ]) 2 = x i,j (1 x i,j )p 2 i,j. Proof: We have E[ˆt 2 i ] (E[ˆt i ]) 2 = V ar[ˆt i ] = x i,j (1 x i,j )p 2 i,j. The first equality follows fro the definition of the variance. The second equality follows fro the linearity of expectation and the independence of the indicator rando variables ˆx ij and ˆx ik for j k. This copletes the proof of the Lea. 6

7 Theore 3.1 The fractional solution to (CQP) can be rounded in polynoial tie to an integral assignent which gives a value which is at ost 2 of the optiu for the l p nor. Proof: We have E[g(ˆt )] = E[ ˆt 2 i ] = E[ˆt 2 i ] = (E[ˆt i ]) 2 + = t 2 i + = f(x, t ) 2OPT 2. x i,j (1 x i,j )p 2 i,j t 2 i + x i,j (1 x i,j )p 2 i,j x i,j p 2 i,j The second equality follows fro the linearity of expectation. The third equality follows fro Lea 3.1 and the last inequality follows fro Lea 2.1. Hence This copletes the proof. (E[g(ˆt )]) 1 2 2OPT. The randoized rounding algorith can easily be derandoized by the ethod of conditional probabilities. 4 Slightly Better Approxiation Algorith for the Restricted Assignent Model (p > 1) Now we turn to the restricted assignent odel. For iproving the 2-Approxiation algorith for the restricted assignent odel we can use either the approxiation algorith presented in the previous section for unrelated achines with its rounding schee or the 2-Approxiation algorith presented by Azar et al. [3] with its rounding schee. When we use the first algorith we denote by H 1 the schedule that contains all the jobs assigned to each achine except the biggest job assigned to each achine. We denote by H 2 the schedule consisting of the big jobs that are not in H 1. When using the latter algorith and its rounding schee which has two phases, we denote by H 1 the schedule consisting of the jobs assigned in the first rounding phase and we denote by H 2 the schedule consisting of the jobs assigned in the second rounding phase. This schedule assigns only one job per achine. We denote by OPT(H 1 ) the optial schedule of the jobs in H 1. We denote by OPT the optial schedule. We have: OPT p H 1 p, (1) OPT p H 2 p, (2) OPT p OPT(H 1 ) p + H 2 p. (3) Now we apply the 2-approxiation algorith for the jobs in H 1, which returns a new schedule H 3. We define the schedule H returned by the algorith as follows: { H1 + H H = 2 H 1 + H 2 p H 2 + H 3 p H 2 + H 3 otherwise 7

8 Next we prove the approxiation ratio of the approxiation algorith. Let ǫ > 0 to be deterined. We consider two cases: If H 2 p (1 ǫ) OPT p then H p H 1 + H 2 p 2 p 1 ( H 1 p + H 2 p ) 2 p 1 ( OPT p + (1 ǫ) OPT p ) = 2 p 1 (2 ǫ) OPT p, where the first inequality follows fro the inequality (x + y) p 2 p 1 (x p + y p ). If H 2 p > (1 ǫ) OPT p we obtain the following: It follows fro (3) that OPT(H 1 ) p ǫ OPT p, hence We obtain H 3 p 2 p OPT(H 1 ) p 2 p ǫ OPT p. H p H 2 + H 3 p 2 p 1 ( H 2 p + H 3 p ) 2 p 1 ( OPT p + 2 p ǫ OPT p ) = 2 p 1 (1 + 2 p ǫ) OPT p. We choose ǫ = 2 p+1, which gives { H p ax 2 p 1 (2 1 } 2 p+1),2p 13 OPT p = (2 p ) OPT p. Hence The following theore suarize the result. H OPT 2(1 1 4p2 p) = 2 1 2p2 p. Theore 4.1 For the restricted assignent odel there is a 2 1 2p2 p approxiation algorith for the l p nor that runs in polynoial tie. 5 PTAS for any fixed nuber of achines and a given l p nor We describe a polynoial tie approxiation schee for any fixed nuber of achines and a given l p nor, i.e. (1 + ǫ)-approxiation algorith for any ǫ > 0 running in polynoial tie. The running tie of the algorith will be bounded by a function that is the product of (n + 1) 2 /ǫ and a polynoial in the size of the input. By the hardness of approxiation result presented in [3], there is no approxiation schee (PTAS or FPTAS) for a given nor and any nuber of achines unless P = NP. Azar et al. [3] showed a fully polynoial tie approxiation schee which is a odification of the ethod presented initially by Horowitz and Sahni in [13]. Our PTAS is a odification of the algorith constructed by Lenstra et. al [14]. The significance of the new algorith is the iproveent in space usage. The space required by the old schee is (n + 1) /ǫ whereas the new schee uses space that is polynoial in both 1 ǫ and (and the input size). For any ǫ > 0 our algorith is as follows: We consider a scheduling proble in the unrelated achines odel, when there is a fixed processing tie p ij associated with each achine i = 1,..., and each job j = 1,...,n. We consider the decision version of the proble with l p nor at ost T. 8

9 For any schedule for the instance with value T, we classify the assignent of a job to a achine as either long or short, depending on whether or not the processing tie in question is greater then ǫt/. No achine can handle /ǫ or ore long assignents before tie T. Thus, for any instance there are less than (n + 1) 2 /ǫ schedules of long assignents. Consider an instance with value T that has a feasible schedule. Let t i be the total processing tie on achine i for that schedule. This schedule includes a partial schedule of long assignents. Suppose that for achine i the long assignents aount to a total processing tie l i, and thus the reaining jobs are copleted within tie d i = t i l i. We define t = ǫt/ For assigning the reaining jobs we solve the following convex progra in t p i x ij 1 for j = 1,...,n x ij p ij t i + l i 0 for i = 1,..., x ij 0 for j = 1,...,n, i = 1,..., x ij = 0 if p ij > t j = 1,...,n, i = 1,..., We present the following theore, which is siilar to a theore given in [16] and has the sae proof. Theore 5.1 If the convex progra has a feasible solution with value less then or equals to T p, then there exists a schedule with l p nor at ost T, such that each achine i has load of at ost t i + t. We see that the convex progra ust have a feasible solution, so that we can apply theore 5.1. The resulting integral solution yields a schedule of short assignents such that the total processing tie taken by short assignents to achine i is at ost t i l i + ǫt/. Cobining this with the schedule of long assignents, we get a schedule where the total tie used by achine i is at ost l i + t i l i + ǫt/ = t i + ǫt/. For the l p nor we obtain t i + ǫt/ t i + ǫt/ T + ǫt = (1 + ǫ)t. We try all possible schedules of long assignents. For each schedule of long assignents we solve the convex progra and apply theore 5.1. If we obtained a schedule using theore 5.1, we return this schedule which has l p nor at ost (1+ǫ)T. Otherwise we answer no. The following theore gives the result. Theore 5.2 The described algorith is a PTAS that requires tie bounded by a polynoial in, log 1/ǫ, and the input size. 6 Approxiation Schee for Multidiensional Jobs (p > 1) We generalize the proble, by defining ultidiensional jobs. In the new proble we consider scheduling parallel unrelated achines. Each job is d-diensional and has to be processed by exactly one achine. Processing job j on achine i in diension k requires tie p ikj. The goal is to find a schedule that iniizes the l p nor of the generalized load vector, which is a atrix where the rows represent the achines and the coluns represent the jobs diension. 9

10 6.1 Convex Prograing Forulation Integer solutions to the following convex progra (CP3) give the optial schedules, where the value of the l p nor of the optial schedule is T 1 p. inf(t ) = d k=1 t p ik x ij 1 for j = 1,...,n x ij p ikj t ik 0 for i = 1,...,, k = 1,...,d c(x ) = d k=1 x ij p p ikj T x ij 0 for j = 1,...,n, i = 1,..., We call the function c(x ) the cost function. 6.2 Convex Prograing Rounding Theore 6.1 If (CP3) has a feasible solution with l p nor value at ost T 1 p, then the fractional solution to (CP3) can be rounded in polynoial tie to an integral assignent which gives a value which is at ost (d + 1)T 1 p for the l p nor. Proof: Given the fractional assignent {x ij } we will show how to construct the desired integral assignent {ˆx ij } in polynoial tie. We ove to the 1-diensional proble by defining 1-diensional d d jobs p ij = p ikj. The load of achine i denoted by t i is t i = t ik. Now for the new 1- k=1 diensional proble instance, we use the fractional solution obtained for the d-diensional instance and we perfor the sae rounding procedure we used for the 1-diensional case, but now We define c ij = d k=1 pp ikj. Then we obtain d x ij c ij = x ij which is the cost function c(x ). After perforing the rounding we return to the original d-diensional jobs. Let {ˆt ik } be the load of achine i in diension k after the rounding, let {ˆq ik } be the processing tie in diension k of the largest job that was atched to achine i by the integral assignent {ˆx ij } and let t ik = ˆt ik ˆq ik. We obtain the following: k=1 k=1 p p ikj ˆt p = t + ˆq p t p + ˆq p d t p + ˆq p Let t i = = d t p + ˆq p dt 1 p + T 1 p = (d + 1)T 1 p. d t ik. The second inequality follows fro the fact that after the rounding t i t i, the k=1 pigeonhole principle and the convexity of the l p nor. The last inequality follows fro the fact that f(t ) T, c(x ) T and the fact that c(ˆx ) c(x ). This copletes the proof. 10

11 References [1] N. Alon, Y. Azar, G. Woeginger, and T. Yadid. Approxiation schees for scheduling on parallel achines. Journal of Scheduling, 1(1):55 66, [2] B. Awerbuch, Y. Azar, E. Grove, M. Kao, P. Krishnan, and J. Vitter. Load balancing in the l p nor. In Proc. of the 36th Annual Syposiu on Foundations of Coputer Science (FOCS95), pages , [3] Y. Azar, L. Epstein, Y. Richter, and G. Woeginger. All-nor approxiation algoriths. In Proc. 8th Scandinavian Workshop on Algorith Theory (SWAT), pages , [4] S. Chakrabarti, C. A. Phillips, A. S. Schulz, D. B. Shoys, C. Stein, and J. Wein. Iproved scheduling algoriths for insu criteria. In Proc. 23rd International Colloquiu on Autoata, Languages and Prograing, pages , [5] F.A. Chudak and D.B. Shoys. Approxiation algoriths for precedence constrained scheduling probles on parallel achines that run at different speeds. Journal of Algoriths, 30: , [6] L. Epstein and J. Sgall. Approxiation schees for scheduling on uniforly related and identical parallel achines. In Proc. 7th Annual European Syposiu on Algoriths, pages , [7] M.X. Goeans and D.P Williason. Iproved approxiation algoriths for axiu cut and satisfiability probles using seidefinite prograing. Journal of the Association for Coputing Machinery, 42: , [8] M. Grötschel, L. Lovasz, and A. Schrijver. The ellipsoid ethod and its consequences in cobinatorial optiization. Cobinatorica, 1: , (Corrigendu:4(1984), ). [9] M. Grötschel, L. Lovasz, and A. Schrijver. Geoetric Algoriths and Cobinatorial Optiization. Springer-Verlag, [10] L. Hall, A. Schulz, D. Shoys, and J. Wein. Scheduling to iniize average copletion tie: Off-line and on-line approxiation algoriths. Matheatics of Operations Research, 22: , [11] D. Hochbau and D. Shoys. A polynoial approxiation schee for scheduling on unifor processors: Using the dual approxiation approach. SIAM Journal on Coputing, 17(3): , [12] D. S. Hochbau and D. B. Shoys. Using dual approxiation algoriths for scheduling probles: Theoretical and practical results. J. Assoc. Coput. Mach., 34(1): , January [13] E. Horowitz and S. Sahni. Exact and approxiate algoriths for scheduling non-identical processors. Journal of the Association for Coputing Machinery, 23: , [14] J.K. Lenstra, D.B. Shoys, and E. Tardos. Approxiation algoriths for scheduling unrelated parallel achines. Math. Prog., 46: , [15] Y. Nesterov and A. Neirovskii. Interior-point polynoial algoriths in convex prograing. SIAM Studies in Applied Matheatics. SIAM,

12 [16] D. Shoys and E. Tardos. An approxiation algorith for the generalized assignent proble. Matheatical Prograing A, 62: , Also in the proceeding of the 4th Annual ACM-SIAM Syposiu on Discrete Algoriths, [17] M. Skutella. Convex quadratic and seidefinite prograing relaxations in scheduling. Journal of the Association for Coputing Machinery, 48(2): , A Appendix A.1 Proof of Theore 2.2 Let. We construct proble instance for the identical achines odel. We consider the achines as points in the interval (0,1], each achine is represented by a point t (0,1], and the load of the achines is represented as a function f(t) in that interval. Let 0 < α < 1. We consider the following instance. There are infinitesially sall jobs of total volue 1 α and unit jobs of total volue α. The optial schedule has l p nor 1 and there is no better fractional assignent. The optial algorith assigns the unit jobs of total volue α evenly to α achines and assigns the infinitesially sall jobs of total volue 1 α evenly to the other 1 α achines. Suppose that the fractional schedule is as follows: The infinitesially sall jobs of total volue 1 α are assigned evenly to all the achines and the unit jobs of total volue α are also assigned evenly to all the achines (α fraction of a unit job is assigned to each achine). Rounding this fractional solution gives the following schedule: The infinitesially sall jobs of total volue 1 α are assigned evenly to all the achines and the unit jobs of total volue α are assigned evenly to α achines (one unit job is assigned to each of these achines). Let A and Opt be the l p nors of the approxiation and off-line algoriths respectively and let C be the approxiation ratio of the algorith. A p = α(2 α) p + (1 α)(1 α) p = α(2 α) p + (1 α) p+1 Opt p = α1 p + (1 α)1 p = 1 ( ) A p C p = α(2 α) p + (1 α) p+1 Opt We choose α = 1 p and we obtain C p 1 p ( 2 p) 1 p ( ) p+1 1 ( 2 1 p. p p p) Hence C ( )1 ( 1 p 2 1 ) ( = e ln p p 2 1 ) ( 1 ln p )( 2 1 ) = 2 O p p p p p ( ln p p ), where the second inequality follows fro the inequality e x 1 x. This copletes the proof. 12