We consider the scheduling model with two identical machines and jobs which arrive online in a list and are assigned to the machines with the objective of minimizing the makespan. Differently from the pure online version, we know in advance a lower bound and also an upper bound on the total size of all jobs. Our algorithm improves previous results on some interval of r, where r is the ratio of the upper and lower bounds on the total size. For most ranges of r our algorithm is best possible. Our technique is based on a smart application of so-called “safe sets”.

In this paper, we consider four single-machine scheduling problems with release times, with the aim of minimizing the maximum lateness. In the first problem we have a common deadline for all the jobs. The second problem looks for the Pareto frontier with respect to the two objective functions maximum lateness and makespan. The third problem is associated with a non-availability constraint. In the fourth one, the non-availability interval is related to the operator who is organizing the execution of jobs on the machine (no job can start, and neither can complete during the operator non-availability period). For each of the four problems, we establish the existence of a polynomial time approximation scheme.

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total flow time. This problem is well known to be NP-complete, and the best polynomial-time approximation algorithms constructed so far had (more or less trivial) worst-case performance guarantees of O(n). In this paper, we present one positive and one negative result on polynomial-time approximations for the minimum total flow time problem: The positive result is the first approximation algorithm with a sublinear worst-case performance guarantee of $O(\sqrt{n})$. This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule. The performance guarantee of our approximation algorithm is not far from best possible, as our second, negative result demonstrates: Unless P=NP, no polynomial-time approximation algorithm for minimum total flow time can have a worst-case performance guarantee of $O(n^{1/2-\eps})$ for any $\eps>0$.

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total ??flow time??. This problem is well-??known to be NP??-complete, and the best polynomial time approximation algorithms constructed so far had (more or less trivial)?? worst-??case performance guarantees of O??(n).????\nIn this paper, we present one positive and one negative result on polynomial time approximations for the minimum total ??flow time problem??. The positive result is the first approxima??tion algorithm with a sublinear worst-??case performance guarantee of O(\\sqrt{n}). This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule??. The performance guarantee of our approximation algorithm is not far from best possible as our second, negative result demonstrates.?? Unless P=NP, no polynomial time approxima??tion algorithm for minimum total ??flow time can have a worst-??case performance guarantee of O(n^{1/2 - \\epsilon}) for any \\epsilon > 0.\n ?? ?? ????\nKeywords:?? scheduling, approximation algorithm, worst-??case analysis, total flow time, release time, single machine??.