On semi-online machine scheduling and generalized bin covering

In this thesis we study algorithms for scheduling problems. We investigate semi-online minimum makespan scheduling and generalized bin covering. In online minimum makespan scheduling we are given a set of m machines and n jobs, where each job Jt is specified by a processing time. The jobs arrive one by one and we have to assign them to the machines without any knowledge about future incoming jobs. The load of a machine is defined to be total processing time of the assigned jobs. The goal is to place the jobs on the machines such that the maximum load of a machine is minimized. In semi-online minimum makespan scheduling this strict setting is softened. We investigate three different models. In the first setting an algorithm is given an advice on the total processing time of the jobs. We present a simple 1.75-competitive algorithm and a lower bound of ≈ 1.585. In the second setting we may reassign jobs upto a limited amount. In this model we present an algorithm that has a competitive ratio of ρm ≈ 1.4659 form→∞ and uses no more than φm ·m job migrations, where φm is a constant between 7 and 10, depending onm. The result is complemented by two lower bounds. Firstly, no algorithm can attain a competitive ratio smaller than ρm using o(n) migrations. Secondly, every algorithm that has competitiveness smaller than 1 + 1/ √ 2 must use Ω(m) job migrations. We finally trade performance for migrations and obtain a family of algorithms with competitiveness ρ̂, for every 5/3 ≤ ρ̂ ≤ 1.75, that uses between 4m and 2.5m job migrations. The third semi-online setting we study is minimum makespan scheduling with parallel schedules. In this problem an algorithm may maintain several schedules, the best of which is output after the arrival of the entire job sequence. We provide a reduction to the special case, where the optimum solution value is known. We design an algorithm that maintains O(1) schedules and is (4/3 + ε)-competitive, for any ε > 0. We furthermore give an algorithm that is (1 + ε)-competitive, for any ε > 0, using mO(1/ε) schedules. Our results are complemented by a lower bound stating that mΩ(1/ε) schedules have to be maintained in order to achieve a competitiveness of less than 1 + ε, for every ε ≤ 1/3. In generalized bin covering we are given m bin types and n items. Each bin type Mj is specified by a demand dj and a revenue rj . Each item Jt has a size pj . A bin of typeMj is said to be covered if the total size of the assigned items is at least the demand dj . Then the revenue rj is earned. The goal is to find an assignment of items to bins maximizing the total obtained revenue. We study two models of bin supply. In the unit supply model there is only one bin of each type available. By contrast in the infinite supply model each bin type is available arbitrarily often, and hence the former is a generalization of the latter. We give a 5-approximation for the unit supply model. In the special case that we have dj = rj , for all bin typesMj , we can give a 9/4-approximation for the unit supply model. We show a lower bound on the approximation factor of 2, unless P = NP, that holds for the unit supply model. The result remains valid even asymptotically. For the infinite supply model we give an AFPTAS in the special case that dj = rj , for all bin types Mj .