Scheduling jobs on identical parallel machines in .NET Attach Data Matrix in .NET Scheduling jobs on identical parallel machines

Scheduling jobs on identical parallel machines generate, create ecc200 none on .net projects .NET Development there is a r .net vs 2010 Data Matrix 2d barcode ounded long job of size greater than T , then there is no such schedule. Otherwise, we can describe an input by a k 2 -dimensional vector, where the ith component speci es the number of long jobs of rounded size equal to iT /k 2 , for each i = 1, .

. . , k 2 .

(In fact, we know that for i < k, there are no such jobs, since that would imply that their original processing 2 requirement was less than T /k, and hence not long.) So there are at most nk distinct inputs a polynomial number! How many distinct ways are there to feasibly assign long jobs to one machine Each rounded long job still has processing time at least T /k. Hence, at most k jobs are assigned to one machine.

Again, an assignment to one machine can be described by a k 2 -dimensional vector, where again the ith component speci es the number of long jobs of rounded size equal to iT /k 2 that are assigned to that machine. Consider the vector (s1 , s2 , . .

. , sk2 ); we shall call it a machine con guration if k2 si iT /k 2 T..

Let C denote the set of all machine con gurations. Note that there are at most (k +1)k distinct con gurations, since each machine must process a number of rounded long jobs that is in the set {0, 1, . .

. , k}. Since k is xed, this means that there are a constant number of con gurations.

Let OPT(n1 , . . .

, nk2 ) denote the minimum number of (identical) machines su cient to schedule this arbitrary input. This value is governed by the following recurrence relation, based on the idea that a schedule consists of assigning some jobs to one machine, and then using as few machines as possible for the rest: OPT(n1 , . .

. , nk2 ) = 1 + min. (s1 ,...,sk2 ) C OPT(n1 s1 , . . .

, nk2 sk2 ).. This can be ECC200 for .NET viewed as a table with a polynomial number of entries (one for each possible input type), and to compute each entry, we need to nd the minimum over a constant number of previously computed values. The desired schedule exists exactly when the corresponding optimal value is at most m.

Finally, we need to show that we can convert the family of algorithms {Bk } into a polynomialtime approximation scheme. Fix the relative error > 0. We use a bisection search procedure to determine a suitable choice of the target value T (which is required as part of the input for each algorithm Bk ).

We know that the optimal makespan for our scheduling input is within the interval [L0 , U0 ], where n L0 = max pj /m , max pj j=1,...

,n j=1 and . U0 = pj /m + max pj . j=1,..

.,n j=1 (We strengthen the lower bound with the by relying on the fact that the processing requirements are integral.) Throughout the bisection search, we maintain such an interval [L, U ], with the algorithmic invariants (1) that L Cmax , and (2) that we can compute a schedule with makespan at most (1 + )U .

This is clearly true initially: by the arguments of Section 2.3, L0 is a lower bound on the length of an optimal schedule, and using a list scheduling algorithm we can compute a schedule of length at most U0 . Electronic web edition.

Copyright 2010 by David P. Williamson and David B. Shmoys.

To be published by Cambridge University Press.
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