Machine scheduling and re - optimization ( extended abstract )

We present approximation algorithms for various machine scheduling problems in the reoptimization setting, which can be described as follows: considering an instance I of a given problem Π for which an optimal solution OPT is provided, and an instance I ′ which results from a local perturbation of I, can the information provided by OPT be used to solve I ′ in a more e cient way (i.e., with a lower complexity and/or with a better approximation ratio) than if this information wasn't available? Although the rst results regarding re-optimization were provided on a parallel machines scheduling problem with forbidden sets in [2], since then, the relevant literature was mainly devoted to graph problems rather than attacking known paradigmatic NP − Hard machine scheduling problems. This is however somehow in contrast with the important literature available on reactive scheduling (see [3]), that is what refers to the schedule modi cations that may have to be made during project execution starting from a baseline schedule. In some cases, the reactive scheduling e ort may rely on very simple techniques aimed at a quick schedule consistency restoration. To this purpose, by considering an optimal schedule as baseline schedule and some jobs additions or deletions as schedule disruptions, we can see that scheduling re-optimization may be considered as strictly linked to reactive scheduling, particularly with simple reoptimization strategies so that the baseline schedule is only mildly modi ed. Here, we consider the re-optimization versions of various strongly NP − Hard machine scheduling problems, under modi cation of the job set (insertion or deletion of jobs). We analyze the approximation ratios of simple re-optimization strategies that keep unchanged the baseline schedule, namely that merely insert the new job (jobs) into the initial optimum under job insertion, or remove the deleted job (jobs) from the initial optimum under job deletion. We denote by Ai,j (1 6 i 6 j 6 n + 1) the reoptimization strategy that consists of computing j − i + 1 candidate solutions, by inserting the new job in the initial optimum in the kth position for all possible k's between i and j, while leaving the rest of the scheduling unchanged. Yet, despite the extreme simplicity of such classes of algorithms, we show that the proposed strategies ensure constant approximation ratios, and also provide tight lower bounds for some of them. As a simple example, consider the F3 || Cmax problem (according to the threeeld notation of [1]) that can be stated as follows. A set of n jobs is available at time 0 to be processed on 3 machines where each job is processed on machine M1 rst, then on machine M2 and nally on machine M3. We denote by pk,j the processing time of job j on machine k (j = 1, ..., n, k = 1, ..., 3). Also, we denote by Ck,j the completion time of job j on machine Mk. Preemption on either machine is not allowed. The objective is the minimization of the maximum completion time on the third machine, that is the makespan. The solution is required to be a permutation schedule, that is the jobs must share the same sequence on all machines. This problem is known to be NP -hard in the strong sense. We denote by F3 || Cmax+ the re-optimization of version