Machine scheduling is a fundamental optimization problem in computer science. The task of scheduling a set of jobs on a given number of machines and minimizing the makespan is well studied and among other results, we know that EPTAS's for machine scheduling on identical machines exist. Das and Wiese initiated the research on a generalization of makespan minimization, that includes so called bag-constraints. In this variation of machine scheduling the given set of jobs is partitioned into subsets, so called bags. Given this partition a schedule is only considered feasible when on any machine there is at most one job from each bag.Das and Wiese showed that this variant of machine scheduling admits a PTAS. We will improve on this result by giving the first EPTAS for the machine scheduling problem with bagconstraints. We achieve this result by using new insights on this problem and restrictions given by the bag-constraints. We show that, to gain an approximate solution, we can relax the bagconstraints and ignore some of the restrictions. Our EPTAS uses a new instance transformation that will allow us to schedule large and small jobs independently of each other for a majority of bags. We also show that it is sufficient to respect the bag-constraint only among a constant number of bags, when scheduling large jobs. With these observations our algorithm will allow for some conflicts when computing a schedule and we show how to repair the schedule in polynomialtime by swapping certain jobs around. ACM Subject Classification Theory of computation → Scheduling algorithms

Online algorithms that allow a small amount of migration or recourse have been intensively studied in the last years. They are essential in the design of competitive algorithms for dynamic problems, where objects can also depart from the instance. In this work, we give a general framework to obtain so called robust online algorithms for these dynamic problems: these online algorithm achieve an asymptotic competitive ratio of γ + ǫ with migration O(1/ǫ), where γ is the best known offline asymptotic approximation ratio. In order to use our framework, one only needs to construct a suitable online algorithm for the static online case, where items never depart. To show the usefulness of our approach, we improve upon the best known robust algorithms for the dynamic versions of generalizations of Strip Packing and Bin Packing, including the first robust algorithm for general d-dimensional Bin Packing and Vector Packing. ACM Subject Classification Theory of computation → Online algorithms

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