A Dynamic Programming Framework for Non-Preemptive Scheduling - PDF document

A Dynamic Programming Framework for Non-Preemptive Scheduling Problems on Multiple Machines [Extended Abstract] Sungjin Im ∗ Shi Li † Benjamin Moseley ‡ Eric Torng § Abstract min-sum objectives, we give the first O (1) -speed O (1) - approximation algorithms for the multiple-machine set- In this paper, we consider a variety of scheduling prob- ting. Even for the single machine case, we reduce both lems where n jobs with release times are to be sched- the resource augmentation required and the approxima- uled non-preemptively on a set of m identical ma- tion ratios. In particular, our approximation ratios are ei- chines. The problems considered are machine minimiza- ther 1 or 1 + ǫ . Most of our algorithms use speed 1 + ǫ tion, (weighted) throughput maximization and min-sum or 2 + ǫ . We also resolve an open question (albeit with a objectives such as (weighted) flow time and (weighted) quasi-polynomial time algorithm) of whether less than 2 - tardiness. speed could be used to achieve an O (1) -approximation for We develop a novel quasi-polynomial time dynamic flow time. New techniques are needed to address this open programming framework that gives O (1) -speed O (1) - question since it was proven that previous techniques are approximation algorithms for the offline versions of ma- insufficient. We answer this open question by giving an chine minimization and min-sum problems. For the algorithm that achieves a (1 + ǫ ) -speed 1 -approximation weighted throughput problem, the framework gives a (1+ for flow time and (1+ ǫ ) -speed (1+ ǫ ) -approximation for ǫ ) -speed (1 − ǫ ) -approximation algorithm. The generic weighted flow time. DP is based on improving a na¨ ıve exponential time DP For the machine minimization problem, we give the by developing a sketching scheme that compactly and ac- first result using constant resource augmentation by show- curately approximates parameters used in the DP states. ing a (1 + ǫ ) -speed 2 -approximation, and the first re- We show that the loss of information due to the sketch- sult only using speed augmentation and no additional ma- ing scheme can be offset with limited resource augmen- chines by showing a (2 + ǫ ) -speed 1 -approximation. We tation.This framework is powerful and flexible, allowing complement our positive results for machine minimiza- us to apply it to this wide range of scheduling objectives tion by considering the discrete variant of the problem and settings. We also provide new insight into the relative and show that no algorithm can use speed augmentation power of speed augmentation versus machine augmen- less than 2 log 1 − ǫ n and achieve approximation less than tation for non-preemptive scheduling problems; specifi- O (log log n ) for any constant ǫ > 0 unless NP admits cally, we give new evidence for the power and importance quasi-polynomial time optimal algorithms. Thus, our re- of extra speed for some non-preemptive scheduling prob- sults show a stark contrast between the two settings. In lems. one, constant speed augmentation is sufficient whereas in This novel DP framework leads to many new algo- the other, speed augmentation is essentially not effective. rithms with improved results that solve many open prob- lems, albeit with quasi-polynomial running times. We highlight our results as follows. For the problems with ∗ Electrical Engineering and Computer Science, Univer- sity of California, 5200 N. Lake Road, Merced CA 95344. sim3@ucmerced.edu . Partially supported by NSF grants CCF- 1008065 and 1409130. Part of this work was done when the author was at Duke University. † Toyota Technological Institute at Chicago, 6045 S. Kenwood Ave. Chicago, IL 606371. shili@ttic.edu . ‡ Department of Computer Science and Engineering, Washing- ton University in St. Louis, St. Louis MO, 63130, USA. bmoseley@wustl.edu . § Department of Computer Science and Engineering, Michigan State University, East Lansing, MI 48824. torng@msu.edu .

( Tar ( WTar ) is harder than FT ( WFT ) since by setting 1 Introduction d J = r J , the Tar ( WTar ) problem becomes the FT ( WFT ) In this paper, we present a new dynamic programming problem). For MM , randomized rounding [24] leads to framework that provides new effective algorithms for an O (log n/ log log n ) -approximation. The approxima- a wide variety of important non-preemptive scheduling tion ratio gets better as opt gets larger. This was the problems. For a typical problem that we study, the input best known algorithm until a breakthrough of Chuzhoy instance consists of a set of n jobs that arrive over time. et al. [11] that showed an O ( opt ) -approximation, where In all but the machine minimization problem, we are also opt is the optimum number of machines needed. That given the number m of identical machines on which we is, the algorithm uses O ( opt 2 ) machines. This implies can schedule jobs. Each job J has a release (or arrival) an O (1) -approximation when opt = O (1) . Combin- time r J , a processing time p J and, depending on the exact ing this and the randomized rounding algorithm gives problem definition, may have a deadline d J or a weight � an O ( log n/ log log n ) -approximation. This is cur- w J . In the unweighted version of a problem, all jobs have rently the best known result for this problem (the O (1) - weight 1. When a job J is scheduled, it must be scheduled approximation result [9] is unfortunately incorrect [10]). for p J consecutive time steps after r J on a machine. Let For Thr and WThr , several Ω(1) approximations are C J be the completion time of job J under some schedule. known [5, 13]. (We use the convention that approxima- The flow time of job J is defined to be F J = C J − r J . tion ratios for maximization problems are at most 1.) In Using our dynamic programming framework, we develop particular, the best approximation ratio for both problems new algorithms and results for the following collection of is 1 − 1 /e − ǫ . problems. If the short name of a problem starts with W , Given the strong lower bounds, particularly for the then jobs have weights. min-sum objective problems and MM , we are forced to • Machine Minimization ( MM ): Jobs have deadlines relax the problem to derive practically meaningful results. and no weights. The goal is to schedule all jobs One popular method for doing this is to use resource aug- by their deadline using the minimum number of mentation analysis where the algorithm is given more re- machines. sources than the optimal solution it is compared against • (Weighted) Throughput Maximization ( WThr , [21]; specifically machine augmentation (extra machines), Thr ): Jobs have deadlines and not all jobs need to be speed augmentation (faster machines), or both machine scheduled. The goal is to maximize the total weight and speed augmentation (extra and faster machines). of the jobs scheduled by their deadline. Bansal et al. [3] applied resource augmentation to most • Total (Weighted) Flow Time ( WFT , FT ): Jobs have of the above problems with m = 1 (or opt = 1 in MM , no deadline. The objective is min � J w J F J . where opt is the optimum number of machines needed). • Total (Weighted) Tardiness ( WTar , Tar ): Jobs Table 1 shows their results. For FT , WFT and Tar , they have deadlines but they do not need to be com- gave 12 -speed 2 -approximation algorithms. For MM and pleted by their deadlines. The objective is Thr , they gave 24 -speed 1 -approximations. Their work in- min � J w J max { ( C J − d J ) , 0 } . troduced an interesting linear program for these problems All of these problems are NP-hard even on a single and rounded the linear program using speed augmenta- machine [15]; NP-hardness holds for the preemptive ver- tion. Their work, unfortunately, does not seem to gen- sions of these problems when we consider multiple ma- eralize to the multiple machine setting even if there are chines. More prior work has been done on the preemptive O (1) machines. We also note that their techniques cannot versions of these problems (see [23, 19] for pointers to be leveraged to obtain O (1) -approximations for the min- some of this work) than the non-preemptive versions. One sum objectives with less than 2 -speed because their linear possible reason for this is the challenge of identifying ef- program has a large integrality gap with less than 2 -speed. fective bounds on the value of the optimal non-preemptive We are motivated by the following open problems, solution for these problems. Finding effective bounds on some more general in nature and others problem specific. the optimal preemptive solution for these problems, while On the general side, we have two main questions. First, also difficult, is easier. One of the key contributions of how can we develop effective lower bounds on the optimal our dynamic programming framework is that we are able solution for a given non-preemptive scheduling instance? to provide effective bounds that allow the development of Second, what is the relative power of speed augmen- approximation algorithms with small approximation ra- tation versus machine augmentation for non-preemptive tios. scheduling problems. On the problem specific side, we Here is a brief summary of prior work on these non- strive to answer the following open questions. Can one preemptive problems. For FT , WFT , Tar and WTar (we use O (1) -speed to get an O (1) -approximation (or even refer to these problems as the min-sum problems), there 1 -approximation) for MM when opt > 1 ? For the min- are very strong lower bounds. Specifically, it is NP- hard to get o ( √ n ) -approximations for these problems [22] sum problems, what can be shown when m > 1 ? Finally,