S. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23:116-127, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....same analysis holds also for load balancing of temporary tasks. However, until now, it was not known whether better approximation algorithms for temporary tasks exist. For the special case of permanent tasks, there is a polynomial time approximation scheme (PTAS) for any xed number of machines [6, 10] and also for arbitrary number of machines by Hochbaum and Shmoys [7] That is, it is possible to obtain a polynomial time (1 ) approximation algorithm for any xed 0. In contrast we show in this paper that the model of load balancing of temporary tasks behaves di erently. Speci cally, for ....

S. Sahni. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery, 23:116-127, 1976. 13

....[53] who use weighted completion time as optimization criterion. With the intractability of many scheduling problems being established, polynomial algorithms guaranteeing a small deviation from the optimal schedule appear more attractive. Some polynomial algorithms are still very complex, [70], while others are particular simple algorithms, like list scheduling methods [41,95] The latter promises to be of the greatest help for the selection of scheduling methods in real systems. Although many of the approximation algorithms have a low computational complexity and produce schedules ....

S. K. Sahni, "Algorithms for scheduling independent tasks". J. ACM 23(1), pp. 116--127, Jan 1976.

....if the number of machines is fixed to m 2. However, for the special case of identical parallel machines Kawaguchi and Kyan [23] showed that list scheduling in order of nonincreasing ratios w j =p j is a 1:21 approximation algorithm. For any fixed number of identical parallel machines Sahni [37] gave a polynomial time approximation scheme which builds upon a general dynamic programming technique of Rothkopf [36] and Lawler and Moore [25] It was observed by Woeginger [48] that the result of Sahni can be extended to a fixed number of uniform parallel machines which run at different but ....

S. Sahni. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery, 23:116 -- 127, 1976.

....all jobs can be scheduled in an instance of 1jj P U j , then EDD nds such a schedule. Moore [24] gave a greedy O(n log n) time optimal algorithm for 1jj P U j . On the other hand, the weighted version 1jj P w j U j is NP hard (knapsack is a special case when all deadlines are equal) Sahni [26] presented a fully polynomial time approximation scheme for this problem. When release dates are introduced, then already 1jr j j P U j is NP hard in the strong sense [13] The following simple greedy rule gives a 2 approximation algorithm: Whenever the machine becomes idle, schedule a job that ....

S. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23:116-127, 1976.

.... improves upon the previously best known #1 # # 2##2 approximation algorithm due to Kawaguchi and Kyan [26] back in the seventies, Sahni gave a fully polynomial time approximation scheme for the problem Pm## w j C j where the number of machines m is constant and does not depend on the input [43]) Subsequently, several research groups have found polynomialtime approximation schemes for problems with release dates such as 1#r j # C j and P # r j # w j C j , the preemptive variant P # r j # pmtn # w j C j , and also for the corresponding problems on a constant number of ....

.... Z # ##1# # # Z # cut by (36) # Z # ##1# # # #m#1# # Z # by (38) 37) # # #m # #1 # # # # Z # # 34 While the problem P## w j C j has a polynomial time approximation scheme [53] even a fully polynomial time approximation scheme when the number of machines is constant [43]) it is shown in [24] that MAXmCUT cannot be approximated within # 1 # 1 34m , unless P=NP. The best currently known approximation algorithms for MAXmCUT have performance ratio 1# 1 m #o # 1 m # which yields #2 # 1 m # approximation algorithms for P## j w j C j by Theorem 6.1. ....

S. Sahni. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery, 23:116 -- 127, 1976.

....jobs can be scheduled in an instance of 1jj P U j , then EDD finds such a schedule. Moore [24] gave a greedy O(n log n) time optimal algorithm for 1jj P U j . On the other hand, the weighted version 1jj P w j U j is NP hard (KNAPSACK is a special case when all deadlines are equal) Sahni [26] presented a fully polynomial time approximation scheme for this problem. When release dates are introduced, then already 1jr j j P U j is NP hard in the strong sense [13] The following simple greedy rule gives a 2 approximation algorithm: When 1 According to the reference, the decision ....

S. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23:116--127, 1976.

....number of packets can arrive per unit time. At each time, the server makes a decision on which packet to serve based on the packets pending in the buffer (i.e. in an on line fashion) The problem of scheduling packets with deadlines has received considerable interest in the literature; e.g. [64, 65, 95, 98, 103, 135, 136, 150]. However, the treatment of on line scheduling problems with multiple classes of traffic remains relatively incomplete. This chapter describes a new family of multiclass scheduling policies, and characterizes its properties and provide substantial empirical results demonstrating significant ....

....that are eligible to be scheduled in the CMTO algorithm; we therefore refer to E t in Step 1 as the eligible set. The computation of E t from P t is a familiar problem in offline scheduling. Indeed, there are several algorithms described in the literature that can be used for this purpose [98, 135, 136, 150]. Here, we describe two new simple example algorithms for Step 1. We present these two new algorithms not as superior alternatives to existing algorithms, but simply to illustrate what is involved in the calculation of Step 1, and show some of the diversity of approaches that are possible. We note ....

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S. Sahni, "Algorithms for scheduling independent tasks," Journal of the ACM , vol. 23, pp. 116--127, 1976.

....problem Pk P c j w j , due to Skutella and Woeginger (1999) Some known results for related problems: 1k P w j c j can be solved in polynomial time using Smith s Ratio Rule. Pk P c j can be solved in polynomial time using SPT Rule. Pmk P w j c j is weakly NP hard and has a FPTAS [1] Rk P c j can be solved in polynomial time. 4] 5] Rk P w j c j does not have a PTAS unless P=NP. P jr j j P w j c j has a PTAS which is also applicable for Rmjr j j P w j c j . Qjr j j P w j c j this problem is still open. 2 The Approximation Scheme 2.1 Approximation for ....

S.Sahni. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery, 23:116-127, 1974.

....(i; j) 2 E is given by w ij = minfw i p j ; p i w j g; the weight n i of the node i 2 V is given by n i = p i . The reason for this setting is, for any single machine scheduling problem with our objective function, the optimal schedule will assign the jobs with the order of nondecreasing p j =w j [11]. Then, the partition (S; V n S) of the node set V can be interpreted as scheduling the n = jV j jobs on the two machines. And P i2S n i T corresponds to the availability constraint that one of the machines is available only in period [0; T ] Moreover, 2 the total weighted completion time of ....

S. Sahni, "Algorithms for Scheduling Independent Tasks," Journal of ACM, 23 (1976) 116127.

....number of packets can arrive per unit time. At each time, the server makes a decision on which packet to serve based on the packets pending in the bu er (i.e. in an online fashion) The problem of scheduling packets with deadlines has received considerable interest in the literature; e.g. [11, 12, 13, 14, 15, 17, 16, 18]. However, the treatment of on line scheduling problems with multiple classes of trac remains relatively incomplete. Our main contribution in this paper is the description of a new family of multiclass scheduling policies, a characterization of its properties, and a rigorous analytical comparison ....

....that are eligible to be scheduled in the CMTO algorithm; we therefore refer to E t in Step 1 as the eligible set. The computation of E t from P t is a familiar problem in o ine scheduling. Indeed, there are several algorithms described in the literature that can be used for this purpose [14, 17, 16, 18]. Here, we describe two new simple example algorithms for Step 1. We present these two new algorithms not as superior alternatives to existing algorithms, but simply to illustrate what is involved in the calculation of Step 1, and show some of the diversity of approaches that are possible. We note ....

[Article contains additional citation context not shown here]

S. Sahni, Algorithms for Scheduling Independent Tasks, J. of the ACM 23 (1976) 116-127.

.... best known (1 p 2) 2 approximation algorithm due to Kawaguchi and Kyan [Kawaguchi and Kyan 1986] back in the seventies, Sahni gave a fully polynomial time approximation scheme for the problem Pm j j P w j C j where the number of machines m is constant and does not depend on the input [Sahni 1976]) Subsequently, several research groups have found polynomial time approximation schemes for problems with release dates such as 1j r j j P C j and P j r j j P w j C j , the preemptive variant P j r j ; pmtn j P w j C j , and also for the corresponding problems on a constant number of ....

.... (1 ) m 1) Z by (38) 37) m (1 ) Z : This completes the proof. While the problem P j j P w j C j has a polynomial time approximation scheme [Skutella and Woeginger 2000] even a fully polynomial time approximation scheme when the number of machines is constant [Sahni 1976]) it is shown in [Kann et al. 1997] that MaxmCut cannot be approximated within 1 1 34m , unless P=NP. The best currently known approximation algorithms for MaxmCut have performance ratio 1 1 m o 1 m which yields (2 1 m ) approximation algorithms for P j j P j w j C j by ....

Sahni, S. 1976. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery 23, 116 - 127.

....the two parallel machines scheduling problem with capacity constraints and our goal is to minimize the total weighted completion time. We will denote this problem by P2=q= P w j C j . Without the capacity constraint, the problem is NP hard but admits a fully polynomial time approximation scheme [19]. We present here an 1:1626approximation algorithm for the P2=q= P w j C j problem with the capacity constraint. 2.2 Progress on algorithm The approximation algorithm is to solve a semide nite programming (SDP) relaxation problem, written in standard form as: Minimize C X (SDP) Subject to ....

S. Sahni, \Algorithms for Scheduling Independent Tasks," Journal of ACM, 23 (1976) 116-127.

....twice the length of the job. Another special case that was considered earlier in the literature is the case in which all jobs are released at the same time (or equivalently, the case in which all deadlines are the same) This special case remains NP Hard even for a single machine. However, Sahni [24] gave a fully polynomial approximation scheme for this special case. The problems considered here have many applications. Hall and Magazine [17] considered the single machine version of our problem in the context of maximizing the scienti c, military or commercial value of a space mission. This ....

S. Sahni, Algorithms for scheduling independent tasks, Journal of the ACM, Vol. 23, pp. 116-127, 1976.

....has a rather rich history. Below we just provide a synopsis of the history, the reader is referred to a recent paper such as [4] for more detailed discussions. The o#ine non preemptive version of the problem for a single machine is NP hard even when all the jobs are released at the same time [25]; however this special case has a fully polynomial time approximation scheme. The o#ine preemptive version of the scheduling problem was studied by Lawler [17] who found a pseudo polynomial time algorithm, as well as polynomial time algorithms for two important special cases. Kise, Ibaraki and ....

Sahni, S, Algorithms for scheduling independent tasks, JACM 23, 116-127, 1976.

....the two parallel machines scheduling problem with capacity constraints and our goal is to minimize the total weighted completion time. We will denote this problem by P2=q= P w j C j . Without the capacity constraint, the problem is NP hard but admits a fully polynomial time approximation scheme [9]. We present here an 1:1626 approximation algorithm for the P2=q= P w j C j problem with the capacity constraint. 2 Preliminaries In this section, we show the relationship between the P2=q= P w j C j problem and the Max (q; n q) Cut problem. In the Max (q; n q) Cut problem, we are given an ....

S. Sahni, \Algorithms for Scheduling Independent Tasks," Journal of ACM, 23 (1976) 116-127.

....the runtime prediction mechanism from Section 3 can be used to determine the execution time of the M tasks for different 7 numbers of processors, we can use a static scheduling strategy. In particular, we use a greedy approximation algorithm to solve this problem which has been introduced in [31]. Theorem 1 Consider the execution of c M tasks J = fJ 1 ; J c g of a layer W on processor groups P = fP 1 ; P g, each containing p = p= processors. We assume that the M tasks in J are sorted according to their predicted execution times, i.e. T (J 1 ; p ) T (J c ; ....

S. K. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23(1):116--127, 1976.

....2 analysis holds also for load balancing of temporary tasks. However, until now, it was not known whether better approximation algorithms for temporary tasks exist. For the special case of permanent tasks, there is a polynomial time approximation scheme (PTAS) for any fixed number of machines [6, 10] and also for arbitrary number of machines by Hochbaum and Shmoys [7] That is, it is possible to obtain a polynomial time (1 ffl) approximation algorithm for any fixed ffl 0. In contrast we show in this paper that the model of load balancing of temporary tasks behaves differently. ....

S. Sahni. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery, 23:116--127, 1976.

....ion Garey et al. 1977] Graham s Longest Processing Time (LPT) algorithm[Graham, 1969] guarantees to find M such that M (4=3 Gamma 1=3n)M opt where M opt is the optimum makespan. Coffmann et al. 1978] give an algorithm based on techniques from bin packing and improve this to 1:22M opt . Sahni [1976] produced a family of approximation algorithms for any guaranteed performance ffl, whose running time was polynomial in l but exponential in n. More recently, Hochbaum and Shmoys [1988a, 1988b] have developed a family of ffl approximation algorithms which is polynomial in both l and n. ....

Sahni, S. (1976). Algorithms for scheduling independent tasks. J. ACM, 23(1):116--127.

....policy involves finding at each time t a CM schedule starting at t for the live unscheduled tasks at time t.Wegive in Figure 1 the pseudo code for an implementation of a particular on line CM scheduler. The problem at each time t is a special case of the general job sequencing problem solved by Sahni (Sahni 1976). Sahni s approach specializes to a O(d 2 ) time complexity CM policy for our problem the CM policy we sketch in Figure 1 can be implemented to obtain a tighter O(d m) performance because it is optimized for our problem assumptions. 6 Our algorithm greedily schedules each task in the latest ....

Sahni 1976 Sahni, S. Algorithms for Scheduling Independent Tasks. Journal of the ACM 23(1):116--127. 1976.

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S. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23:116-127, 1976.

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S. Sahni. Algorithms for scheduling independent tasks. Journal of ACM, 23:116-- 127, 1976.

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S. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23:116--127, 1976.

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Sartaj K. Sahni, "Algorithms for Scheduling Independent Tasks," Journal of the Association for Computing Machinery, Vol. 23, No. 1, pp. 116--127, 1976.

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S. Sahni. Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery, 23:116--127, 1976.

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S. Sahni. Algorithms for Scheduling Independent Tasks. Journal of the ACM, 23:116--127, 1976.

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