This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.

This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.

The objective of the classical Joint Replenishment Problem (JRP) is to minimize ordering costs by combining orders in two stages, first at some retailers, and then at a warehouse. These orders are needed to satisfy demands that appear over time at the retailers. We investigate the natural special case that each demand has a deadline until when it needs to be satisfied. For this case, we present a randomized 5/3-approximation algorithm. We moreover prove that JRP with deadlines is APX-hard. Finally, we extend the known hardness results by showing that JRP with linear delay cost functions is NP-hard, even if each retailer has to satisfy only three demands.

We consider the classical problem of scheduling preemptible jobs, that arrive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj, pmtn | P j cj . A popular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [13,14]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ratio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [11].

Given an interval graph and integer k, we consider the problem of finding a subgraph of size k with a maximum number of induced edges, called densest k-subgraph problem in interval graphs. It has been shown that this problem is NP-hard even for chordal graphs [17], and there is probably no PTAS for general graphs [12]. However, the exact complexity status for interval graphs is a long-standing open problem [17], and the best known approximation result is a 3-approximation algorithm [16]. We shed light on the approximation complexity of finding a densest k-subgraph in interval graphs by presenting a polynomialtime approximation scheme (PTAS), that is, we show that there is an (1 + ǫ)-approximation algorithm for any ǫ > 0, which is the first such approximation scheme for the densest k-subgraph problem in an important graph class without any further restrictions.