We investigate the scheduling of n jobs divided into c classes on m identical parallel machines. For every class there is a setup time which is required whenever a machine switches from the processing of one class to another class. The objective is to find a schedule that minimizes the makespan. We give near-linear approximation algorithms for the following problem variants: the non-preemptive context where jobs may not be preempted, the preemptive context where jobs may be preempted but not parallelized, as well as the splittable context where jobs may be preempted and parallelized.We present the first algorithm improving the previously best approximation ratio of 2 to a better ratio of 3/2 in the preemptive case. In more detail, for all three flavors we present an approximation ratio 2 with running time O(n), ratio 3/2 + ε in time O(n log 1/ε) as well as a ratio of 3/2. The (3/2)-approximate algorithms have different running times. In the non-preemptive case we get time O(n log(n + ∆)) where ∆ is the largest value of the input. The splittable approximation runs in time O(n + c log(c + m)) whereas the preemptive algorithm has a running time O(n log(c + m)) ≤ O(n log n). So far, no PTAS is known for the preemptive problem without restrictions, so we make progress towards that question. Recently Jansen et al. found an EPTAS for the splittable and nonpreemptive case but with impractical running times exponential in 1/ε.

We study Nonpreemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the strip is minimized. NPDM can be modeled as strip packing with slicing and stacking constraint: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions where two slices of the same rectangle are intersected by the same vertical line. Nonpreemption enforces the slices to be placed in contiguous horizontal locations (but may be placed at different vertical locations).We obtain a (5/3 + ε)-approximation algorithm for the problem. We also provide an asymptotic efficient polynomial-time approximation scheme (AEPTAS) which generates a schedule for almost all jobs with energy consumption (1 + ε)OPT. The remaining jobs fit into a thin container of height 1. The previous best for NPDM was 2.7 approximation based on FFDH [40]. One of our key ideas is providing several new lower bounds on the optimal solution of a geometric packing, which could be useful in other related problems. These lower bounds help us to obtain approximative solutions based on Steinberg's algorithm in many cases. In addition, we show how to split schedules generated by the AEPTAS into few segments and to rearrange the corresponding jobs to insert the thin container mentioned above.

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We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer s that is specified by n fixed remainders modulo integer divisors a1, . . . , an we consider remainder intervals R1, . . . , Rn such that s is feasible if and only if s is congruent to ri modulo ai for some remainder ri in interval Ri for all i.This problem is a special case of a 2-stage integer program with only two variables per constraint which is is closely related to directed Diophantine approximation as well as the mixing set problem. We give a hardness result showing that the problem is NP-hard in general.Motivated by the study of the mixing set problem and a recent result in the field of realtime systems we investigate the case of harmonic divisors, i.e. ai+1/ai is an integer for all i < n. We present an algorithm to decide the feasibility of an instance in time O(n 2 ) and we show that even the smallest feasible solution can be computed in strongly polynomial time O(n 3 ).