Design of Approximation Algorithms

3.2 Scheduling jobs on identical parallel machines 71there is a rounded 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 specifies thenumber of long jobs of rounded size equal to iT/k 2 , for each i = 1, . . . , k 2 . (In fact, we knowthat for i < k, there are no such jobs, since that would imply that their original processingrequirement was less than T/k, and hence not long.) So there are at most n k2 distinct inputs— a polynomial number!How many distinct ways are there to feasibly assign long jobs to one machine? Each roundedlong 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 againthe ith component specifies the number of long jobs of rounded size equal to iT/k 2 that areassigned to that machine. Consider the vector (s 1 , s 2 , . . . , s k 2); we shall call it a machineconfiguration if∑k 2s i · iT/k 2 ≤ T.i=1Let C denote the set of all machine configurations. Note that there are at most (k +1) k2 distinctconfigurations, since each machine must process a number of rounded long jobs that is in theset {0, 1, . . . , k}. Since k is fixed, this means that there are a constant number of configurations.Let OPT(n 1 , . . . , n k 2) denote the minimum number of (identical) machines sufficient toschedule this arbitrary input. This value is governed by the following recurrence relation, basedon the idea that a schedule consists of assigning some jobs to one machine, and then using asfew machines as possible for the rest:OPT(n 1 , . . . , n k 2) = 1 + min OPT(n 1 − s 1 , . . . , n k 2 − s k 2).(s 1 ,...,s k 2 )∈CThis can be viewed as a table with a polynomial number of entries (one for each possibleinput type), and to compute each entry, we need to find the minimum over a constant numberof previously computed values. The desired schedule exists exactly when the correspondingoptimal value is at most m.Finally, we need to show that we can convert the family of algorithms {B k } into a polynomialtimeapproximation scheme. Fix the relative error ϵ > 0. We use a bisection search procedureto determine a suitable choice of the target value T (which is required as part of the input foreach algorithm B k ). We know that the optimal makespan for our scheduling input is withinthe interval [L 0 , U 0 ], where⎧⎡⎤ ⎫⎨ n∑⎬L 0 = max⎩p j /m⎢ ⎥ , max p jj=1,...,n ⎭andj=1⎡ ⎤n∑U 0 =p j /m⎢ ⎥ + max p j.j=1,...,nj=1(We strengthen the lower bound with the ⌈·⌉ by relying on the fact that the processing requirementsare integral.) Throughout the bisection search, we maintain such an interval [L, U], withthe algorithmic invariants (1) that L ≤ C ∗ max, and (2) that we can compute a schedule withmakespan at most (1 + ϵ)U. This is clearly true initially: by the arguments of Section 2.3, L 0is a lower bound on the length of an optimal schedule, and using a list scheduling algorithm wecan compute a schedule of length at most U 0 .Electronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press