SODA Conference 2015 Conference Paper
- Sungjin Im
- Shi Li 0001
- Benjamin Moseley
- Eric Torng
In this paper, we consider a variety of scheduling problems where n jobs with release times are to be scheduled non-preemptively on a set of m identical machines. The problems considered are machine minimization, (weighted) throughput maximization and min-sum objectives such as (weighted) flow time and (weighted) tardiness. We develop a novel quasi-polynomial time dynamic programming framework that gives O (l)-speed O (l)-approximation algorithms for the offline versions of machine minimization and min-sum problems. For the weighted throughput problem, the framework gives a (1 + ε)-speed (1 – ε)-approximation algorithm. The generic DP is based on improving a naïve exponential time DP by developing a sketching scheme that compactly and accurately approximates parameters used in the DP states. We show that the loss of information due to the sketching scheme can be offset with limited resource augmentation. This framework is powerful and flexible, allowing us to apply it to this wide range of scheduling objectives and settings. We also provide new insight into the relative power of speed augmentation versus machine augmentation for non-preemptive scheduling problems; specifically, we give new evidence for the power and importance of extra speed for some non-preemptive scheduling problems. This novel DP framework leads to many new algorithms with improved results that solve many open problems, albeit with quasi-polynomial running times. We highlight our results as follows. For the problems with min-sum objectives, we give the first O (l)-speed O (l)-approximation algorithms for the multiple-machine setting. Even for the single machine case, we reduce both the resource augmentation required and the approximation ratios. In particular, our approximation ratios are either 1 or 1 + ε. Most of our algorithms use speed 1 + e or 2 + ε. We also resolve an open question (albeit with a quasi-polynomial time algorithm) of whether less than 2-speed could be used to achieve an O (1)-approximation for flow time. New techniques are needed to address this open question since it was proven that previous techniques are insufficient. We answer this open question by giving an algorithm that achieves a (1 + ε)-speed 1-approximation for flow time and (1 + ε)-speed (1 + ε)-approximation for weighted flow time. For the machine minimization problem, we give the first result using constant resource augmentation by showing a (1 + ε)-speed 2-approximation, and the first result only using speed augmentation and no additional machines by showing a (2 + ε)-speed 1-approximation. We complement our positive results for machine minimization by considering the discrete variant of the problem and show that no algorithm can use speed augmentation less than 2 log 1–ε and achieve approximation less than O (log log n ) for any constant ε > 0 unless NP admits quasi-polynomial time optimal algorithms. Thus, our results show a stark contrast between the two settings. In one, constant speed augmentation is sufficient whereas in the other, speed augmentation is essentially not effective.