AN IMPROVED 2.11-COMPETITIVE ALGORITHM FOR ONLINE SCHEDULING ON PARALLEL MACHINES TO MINIMIZE TOTAL WEIGHTED COMPLETION TIME

We revisit the classical online scheduling problem on parallel machines for minimizing total weighted completion time. In the problem, a set of independent jobs arriving online over time has to be scheduled on identical machines, where the information of each job including its processing time and weight is not known in advance. The goal is to minimize the total weighted completion time of the jobs. For this problem, we propose an improved 2.11competitive online algorithm based on a kind of waiting strategy.


1.
Introduction. Until two decades ago, one of the basic assumptions made in deterministic scheduling was that all of the information needed to define the problem instance was known in advance. This assumption is usually not valid in practice, however. Abandoning it has led to the rapidly emerging field of on-line scheduling. A simple instance is how to arrange the printing materials that can arrive at any time in a print shop. The introduction of online scheduling greatly enriched the field of scheduling. Since then, volumes of research has focused on design, implementation, and analysis algorithms for an online scheduling problem. In contrast to the off-line version, an online algorithm must produce a sequence of decisions based on past events without any information about the unreleased jobs. The lack of knowledge of the future does not generally guarantee the optimality of the schedule generated by an online algorithm. Thus a natural issue is how to evaluate different online algorithms for a same scheduling problem. A widely used approach to evaluate from a single machine to identical machines for the case that all jobs have identical weights and obtained a 2-competitive online algorithm. In a celebrated paper, Correa and Wagner [3] proposed a 2.618-competitive online algorithm for problem P |online, r j | w j C j by combining the α-point method with linear programming relaxation. In [13], Sitters designed an online algorithm with the competitive ratio not greater than α m = (1+1/ √ m) 2 (3e−2) 2e−2 by utilizing the technique of shifting release times. The value α m is very great when the machine number m is small, although it tends to 1.791 when m tends to infinity.
In a recent study, Tao [15] revisited problem P |online, r j | w j C j and presented an online algorithm AD-SWPT(Average Delayed Shortest Weighted Processing Time). By the technique of instance reduction, they showed that AD-SWPT is (2.5 − 1/2m)-competitive, where m is the number of the machines. Also based on the same instance reduction, by supplying the following algorithm Improved AD-SWPT, Tao et al. [16] improved the above result to (1 + )competitive, which tends to 2.28 when m tends to infinity. Let t be the current decision time,p j (t) the remaining processing time of job J j at time t in an online schedule, S j the starting time of job J j in the online schedule. The improved AD-SWPT rule can be described in detail as follows: Improved AD-SWPT: Whenever there is one idle machine and some jobs are available, choose a job with the smallest Smith-ratio p j /w j among all the arrived and unscheduled jobs. Suppose that J i is chosen as the candidate job. If ≤ αt, where Sj ≤tp j (t) is the total remaining processing time at all the busy machines at time t, then we schedule J i from t at the idle machine; otherwise, wait until a new job arrives or the above equation satisfies, where α ≥ 1 is a parameter to be designed.
From the description of Improved AD-SWPT, Improved AD-SWPT is an algorithm obtained by pumping a flexible parameter α into AD-SWPT presented in [15], and so Improved AD-SWPT can be reduced to AD-SWPT when parameter α = 1. Thus, the competitive ratio of Improved AD-SWPT should be lower than or equal to that of AD-SWPT.
For the online scheduling problem P |online, r j | w j C j considered in our paper, Tao et al. in [16] provided Improved AD-SWPT algorithm and showed that Improved-ADSWPT is a polynomial-time online algorithm with a competitive ratio of at most 2.28. We improve the above result and show that Improved-ADSWPT is a polynomial-time online algorithm with a competitive ratio of at most 2.11.
Our contribution: For online scheduling problem P |online, r j | w j C j , we establish that Improved AD-SWPT is 2.11-competitive by combining the technique "improved instance reduction" together with the lower bounds provided respectively in Kawaguchi and Kyan [7] and Chou et al. [2]. In this case, though we cannot show that our analysis is tight, according to the computational study provided in [15], the remaining gap is at most 0.11. Finally, we present our conjecture that Improved AD-SWPT has a competitive ratio of 2.
2. Preliminaries. The following notations will be used throughout this paper: • I: the job instance I.
• p j : the processing time of job J j .
• r j : the release date of job J j .
• w j : the weight of job J j .

RAN MA AND JIPING TAO
• τ j = p j /w j : the Smith-ratio of job J j .
• τ max : the maximum Smith-ratio of the jobs.
• (I, σ): an instance-schedule pair if σ is a feasible schedule of the job instance I.
• S j (I, σ): the starting time of job J j in (I, σ).
• C j (I, σ) = S j (I, σ) + p j : the completion time of job J j in (I, σ).
• For a subset T ⊆ I, we use S T (I, σ) and C T (I, σ) to denote the first starting time and the last completion time of the jobs in T in (I, σ), respectively. Especially, when T = I, we write S I (I, σ) and C I (I, σ) for short.
• Z j (I, σ) = w j (I)C j (I, σ): the contribution of job J j ∈ I to the objective value of (I, σ).
• For a subset T ⊆ I, we use Z T (I, σ) = Jj ∈T Z j (I, σ) to denote the total contribution of the jobs in T to Z (I, σ).
• π(I): an optimal (offline) schedule of the job instance I.
• OPT(I): the objective value of an optimal (offline) schedule π(I).
• t: the current decision time.
•p j (t): the remaining processing time of job J j at time t in a feasible schedule.
The following lemmas, which will be repeatedly used in our competitive analysis, are provided without proof, since they can be derived from the basic mathematics and have been frequently used in the literature such as [10,9,15].
reaches its maximum value at one endpoint of the interval, i.e., Lemma 2.2. Let f (I) and g(I) be two positive-valued functions defined on the job instance I. If I, I 1 and I 2 are three instances such that f (I) ≤ f (I 1 ) + f (I 2 ) and Let I be a job instance. For each T ⊆ I and each positive number δ, we use I (T,δ) to denote the job instance obtained from I satisfying that the weight w j (I) of each job J j ∈ T is modified as w j (I (T,δ) ) = δ · w j (I). Since each job J j ∈ I (T,δ) has a release date r j (I (T,δ) ) = r j (I), a feasible schedule of I is also a feasible schedule of I (T,δ) , and vice versa. Denote by σ| T the subschedule of the jobs restricted to T in schedule σ (I). We have the following useful lemmas, which are similar to the lemmas in Ma and Yuan [10]. For the sake of completeness, we provide the lemmas together with the proof. Lemma 2.5. Let X(I (T,δ) ) be a schedule of I (T,δ) which processes each job in the same time interval as its corresponding job in σ(I) for each δ ∈ [u, v]. Then 2.11-COMPETITIVE ALGORITHM FOR IDENTICAL MACHINE SCHEDULING 501 Z(I (T,δ) , X) is a convex function and OPT (I (T,δ) ) is a concave function, respectively, with respect to δ for each δ ∈ [u, v].
Proof. In fact, since X(I (T,δ) ) maintains the starting time of each job with δ changing in [u, v], we have Z(I (T,δ) , X) = Z(I, σ) + (δ − 1) · Jj ∈T Z j (I, σ). Then Z(I (T,δ) , X) is a monotonously increasing linear function and so a convex function with respect to δ for each δ ∈ [u, v].
Let Π be the set of all feasible schedules of instance I. Then Π is also the set of all feasible schedules of instance I (T,δ) . For each θ ∈ Π, we have Z( u, v]. Thus, the lemma follows. Let δ be an arbitrary positive number with δ ≤ 1. Let X(I (T,δ) ) be a schedule of I (T,δ) which processes each job in the same time interval as its corresponding job in Proof. Suppose to the contrary that Z(I (T,δ) , X)/OPT(I (T, δ) ) < Z(I, σ)/OPT (I).
Since Z(I (T , δ) , X) = δ · Z(I (T, δ) , σ) and OPT(I (T , δ) ) = δ · OPT(I (T, δ) ), we also have Z(I (T, δ) , X)/OPT(I (T, δ) ) ≥ Z(I, σ)/OPT (I). The lemma follows. Suppose that (I, σ) be an instance-schedule pair. In what follows of this paper, according to the increasing order of the staring times of jobs in σ(I), all jobs are denoted by J 1 , J 2 , . . . , J n , with ties brocken in nondecreasing Smith-ratio. We call I a regular instance if all jobs in I have a common Smith-ratio τ and there does not exist a time t between the earliest processing time and the latest completion time in online schedule σ (I) such that all the machines remain idle at time t. Now we will give Lemma 2.8 and Lemma 2.9 for a regular instance. As dedicated in the proof, Lemma 2.8 and Lemma 2.9 assert that if I is a regular instance, then we have Though the proof in Lemma 2.8 is similar with [16], we can simplify the complicated deduction in [15] and [16] by taking advantage of the following lower bound in [2] instead of the lower bound from mean-busy-time in [15] and [16].
For problem P m|online, r j | w j C j with p j = w j for each job, from [2], the objective value u(I) of optimal schedules satisfies an inequality of where r min is the earliest release time.
Since I is a regular instance, we can normalize the ratio of p j /w j to 1 by rescaling the weights of all jobs. Consider the time in σ(I), say r L , from which jobs are continuously processed after r L at each machine without idle time between jobs. Now we present Lemma 2.8 and Lemma 2.9 according to two different cases of the jobs processed after r L , respectively. Lemma 2.8. If there does not exist a job which is released before r L and is scheduled at or after r L in σ(I), then Z(I, σ)/OPT(I) ≤ 1 + α.
Proof. Consider these jobs that start at or after r L , also denoted by J s , J s+1 , . . . , J n .
The assumption in the lemma implies that J s , J s+1 , . . . , J n must be released at or after r L , and so these jobs have no effect on the jobs starting before r L . Construct an intermediate instance I 1 = I \ {J s , J s+1 , . . . , J n }. Then we have Note that the jobs J s , J s+1 , . . . , J n are continuously processed after r L at each machine. So we can limit the starting time of the jth job in {J s , J s+1 , . . . , J n } as where the second term is to average the total processing time which has to be finished between r L and S j over all the machines.

2.11-COMPETITIVE ALGORITHM FOR IDENTICAL MACHINE SCHEDULING 503
Let A = Si<r Lp i (r L ), B = n j=s p j . Along with (2) and (3), we can limit Z(I, σ) by an upper bound as Consider the set of {J s , J s+1 , . . . , J n } as a separate instance, and further relax the release times of all the jobs to r L , then we can develop a lower bound on the optimal schedule π(I) according to (1).
According to Improved AD-SWPT, we have Sj ≤r Lp j (r L )/m ≤ αr L , and so A/m ≤ αr L . Combining (4) and (5), together with the fact α ≥ 1, we have Rewrite I 1 as I and repeat the analysis above. Ultimately the performance ratio of I can be bounded by 1 + α. The lemma follows. Lemma 2.9. If there exists at least one job which is released before r L and starts at or after r L in σ(I), then Z(I, σ)/OPT(I) ≤ 1+ According to Improved AD-SWPT, there is a job J k must satisfy Consider these jobs which are completed after r L , also denoted by J s , J s+1 , . . . , J n . Construct an intermediate instance I 1 = I \ {J s , J s+1 , . . . , J n }. Divide the set of {J s , J s+1 , . . . , J n } into two subsets as follows: Let Jj ∈Q1 p j := A, and Jj ∈Q2 p j := B.
For the purpose of our performance analysis, now we take only the jobs in Q 1 ∪Q 2 into consideration and create a new schedule σ * (Q 1 ∪ Q 2 ) of Q 1 ∪ Q 2 as follows: we put off the starting times of jobs in Q 1 \{J k } until time r L , the other jobs in Q 1 ∪ Q 2 are scheduled after time r L according to Improved AD-SWPT, thus the jobs in Q 1 ∪ Q 2 are scheduled in LS rule starting at time r L . Clearly, we have Z Q1∪Q2 (I, σ) ≤ Z(Q 1 ∪ Q 2 , σ * ). Let I be the job instance by resetting the release date as r j = 0 for each J j ∈ Q 1 ∪ Q 2 . Let σ (I ) be a schedule of I starting at time 0 such that the jobs in σ (I ) are scheduled in the same LS rule as in σ * (Q 1 ∪ Q 2 ).
Recall that in a well-known paper, for scheduling problem P m|| w j C j , Kawaguchi and Kyan [7] showed that the total weighted completion time of an LRF (Largest-Ratio-First) schedule is at most a factor 1+ √ 2 2 larger than the optimal total weighted completion time. This means that, for scheduling problem P m|| w j C j , any LS schedule is at most a factor 1+ √ 2 2 larger than the optimal total weighted completion time for an instance in which w j = p j holds for all jobs. We can regard this as a new lower bound of our problem. From (1), we can obtain that OPT (I ) (7), we can derive that A/m ≥ αr L and so r L ≤ A mα . Furthermore we can obtain Jj ∈Q1 p 2 j ≥ A 2 /m because there are at most m jobs in Q 1 . Thus, we have Rewrite I 1 as I and repeat the analysis above. Ultimately the performance ratio of I can be bounded by 1+ α . The lemma follows.
3. Algorithm analysis. In order to apply the approach of "improved instance reduction" correctly, we present the following new online algorithm Flexible Improved AD-SWPT or FIADSWPT in short by pumping into Improved AD-SWPT some flexibility.
FIADSWPT: Whenever there is one idle machine and some jobs are available, choose a job with the smallest Smith-ratio τ j = p j /w j among all the arrived and 2.11-COMPETITIVE ALGORITHM FOR IDENTICAL MACHINE SCHEDULING 505 unscheduled jobs. Suppose J i is chosen as the candidate job. If where Sj ≤tp j (t) is the total remaining processing time at all the busy machines at time t, then we schedule J i from t at the idle machine; otherwise, wait until the next time, where α ≥ 1 is a parameter to be designed.
In FIADSWPT, a candidate job at time t is a job with the smallest Smith-ratio among all the arrived and unscheduled jobs. Moreover, a candidate job J i is called a ready job at time t if pi+ S j ≤tp j (t) m ≤ αt at time t. Due to the flexibility, algorithm FIADSWPT is not polynomial-time; in addition, for a given job instance, FIADSWPT may generate different schedules, each one of them is called a possible schedule generated by FIADSWPT. But, it is the flexibility of FIADSWPT that enables us to use the technique of "improved instance-reduction" freely in the analysis of the competitive ratio of FIADSWPT. We finally will show in this paper that, for a given job instance, each possible schedule generated by FIADSWPT has an objective value at most λ(α) times the objective value of an optimal (off-line) schedule. Note that the schedule generated by Improved AD-SWPT is also a possible schedule generated by FIADSWPT. Then we can conclude that Improved AD-SWPT is a λ(α)-competitive polynomial-time online algorithm for problem P |online, r j | w j C j .
For a job instance I and a schedule σ obtained by FIADSWPT on I, we call (I, σ) an online instance-schedule pair. The following lemma characterizes the online instance-schedule pairs (I, σ). } ≤ t < S j (I, σ) in (I, σ), there is a candidate job J i = J j with the smallest Smith-ratio among all jobs at time t in (I, σ) such that τ i (I) ≤ τ j (I) and It is sufficient to show that Z(I, σ)/OPT(I) ≤ λ(α) for each online instanceschedule pair (I, σ). We will take advantage of the approach of "improved instance reduction" in our analysis. By combining "improved instance reduction" with contradiction, we will search for the instances I with a special structure such that Z(I, σ)/OPT(I) ≥ λ(α) until a specific instance with a more special structure is obtained. In fact, the specific instance is just a regular instance. However, Lemma 2.8 and Lemma 2.9 told us that Z(I, σ)/OPT(I) ≤ λ(α) for any regular instance. Hence, we can come to the conclusion that Z(I, σ)/OPT(I) ≤ λ(α) for each online instance-schedule pair (I, σ).
An instance I is said to be a counterexample if Z(I, σ)/OPT(I) > λ(α) for a certain possible schedule σ(I) generated by FIADSWPT on instance I. Furthermore, I is called a smallest counterexample if I is a counterexample such that the number of jobs in I is as small as possible. For the case that I is a smallest counterexample, we also say that (I, σ) is a smallest violated online instance-schedule pair. We define Ω to be the set of all smallest violated online instance-schedule pairs. Suppose to the contrary that FIADSWPT has a competitive ratio greater than λ(α). Then Ω is not empty.
We call a set of jobs a block if there does not exist a time t in the time interval of the processing of the set of jobs such that all the machines remain idle at t in σ (I). Thus, the jobs in σ(I) are divided by idle intervals into some blocks of σ (I).
We further partition the jobs in each block of σ(I) into subblocks such that within each subblock the jobs are scheduled in SWPT-rule, and that the ratio τ j of the last job of a subblock is larger than that of the first job of the succeeding subblock if it exists. We first establish some lemmas to reveal the properties of (I, σ) in the following.
Lemma 3.2. For each (I, σ) ∈ Ω, there is no job J j in I such that τ j = 0.
Proof. It suffices to show that there is no job J j in I such that p j = 0. Suppose to the contrary that there is some job J j ∈ I with p j = 0. By algorithm FIADSWPT, J j will be processed as soon as one of the machines becomes idle after it is released at time r j . We distinguish the following into two cases: Case 1. J j starts at time r j in σ (I). Then I = I \ {J j } is a counterexample with a smaller number of jobs since Z(I, σ)/OPT(I) ≤ Z(I , σ)/OPT (I ). This contradicts the minimality of I. Case 2. J j is processed after time r j in σ (I). This means that all machines are busy at time r j . Let J i denote the latest processed job before r j and S i the starting time of job J i . Note that S i is the latest starting time before r j , then all machines are busy at time S i . By FIADSWPT, we can obtain that pi+ S j ≤S ip j (Si) m ≤ αS i , and so there must exist a job, say J k , such that S k ≤ r j andp k (S i ) ≤ αS i . Thus, C j (σ(I)) = C k (σ(I)) = S i (σ(I)) +p k (S i ) ≤ (1 + α)S i ≤ (1 + α)r j . Let I = I \ {J j }. Consider an instance-schedule pair (I, σ) ∈ Ω. The last job in a block B of σ(I) is called the end-job of B in σ (I). For each job J k ∈ σ(I), we use B (k) to denote the block in σ(I) including J k .
Let J k ∈ σ(I) which is not the end-job of B (k) . We use J k + to denote the successor job of J k , i.e., J k + ∈ B (k) and there is no job to start between time S k + (σ(I)) and time S k (σ(I)). We call (J k , J k + ) an SWPT-reverse pair in σ(I) if τ k (I) > τ k + (I). In the case that there is no other SWPT-reverse pair (J j , J j + ) in σ(I) with S j (σ(I)) > S k (σ(I)), we also call (J k , J k + ) the last SWPT-reverse pair in σ (I).
For an SWPT-reverse pair (J k , J k + ) in σ(I), the following notations are used in our deduction.