Research Note
A note on labeling methods to schedule unit execution time tasks in the presence of delayed precedence constraints

https://doi.org/10.1016/j.jpdc.2021.05.002Get rights and content

Highlights

  • Addressing the worst-case performance of labeling methods like the Coffman-Graham algorithm.

  • Focus on delayed precedence constraints in the single- and multi-processor case.

  • New lower bounds on worst-case approximation guarantees are derived.

  • The results extend to more general level-based or critical-path scheduling techniques.

  • An open problem by Bernstein, Rodeh, and Gertner from 1989 is resolved.

Abstract

There is some evidence that labeling schemes as employed for instance in the famous Coffman-Graham algorithm may provide superior worst-case approximation guarantees than purely path- or level-based list schedules in the presence of (delayed) precedence constraints. In 1989, Bernstein, Rodeh, and Gertner proved that this also holds true for their labeling scheme targeting unit execution time tasks on a single processor provided that all delays imposed by a single task on all of its successors are uniformly either zero or some fixed positive integer. They further conjectured that this superiority is preserved when allowing the delays imposed by a task to differ successor-wise. It is shown in this note however that their labeling scheme as well as more general ones may perform as bad as any list schedule in this case. Moreover, a new lower bound on the worst-case performance of labeling methods in the multiprocessor setting is derived.

Introduction

Let T be a set of tasks and let pjZ0 denote the non-negative processing or execution time of each task jT. We consider the problem of scheduling the tasks T subject to delayed precedence constraints so as to minimize the makespan. In this setting, we are given a partial order ≺ resembling tasks dependencies each of which is accompanied by a delay or latency. More precisely, if i,jT and ij then the associated delay ijZ0 imposes that if i starts at some time s(i), then j can start no earlier than at time s(i)+pi+ij. We will employ this more general notation whenever applicable although the focus will be on the case of unit execution times (UET), i.e., pj=1 for all jT.

In the three-field-notation proposed in [12], the problem under consideration is denoted as P|pj=1,prec(ij)|Cmax. It is known to be NP-hard if the delays ij can be arbitrary non-negative, even on a single processor [8], [13]. It remains NP-hard even if all delays are fixed to some positive integer,1 and if the precedence relationships form a chain [25] or a bipartite graph [17].

Delayed precedence constraints arise for instance from the pipelined structure of real-world processors implementing one particular form of instruction-level parallelism. Therefore, the problem and algorithms to solve it have been intensively studied in this context (see e.g. [2], [3], [8], [13], [21], [22], [24]), albeit not as intensively as in the related settings without delays or with communication delays.

The major focus in this note is on so-called labeling methods that strongly relate to (or are even derived from) the famous Coffman-Graham algorithm [7] and that are specialized list scheduling techniques to be precisely defined in Sect. 2. One reason for this focus is that there is some evidence for the superiority of labeling methods compared to other list scheduling algorithms as will be discussed more comprehensively in Sect. 3.

As one example that is not entirely resolved so far, Bernstein, Rodeh and Gertner show in [3] that, if each single task can impose only a delay of zero or some fixed integer L1 on all of its successors, then the makespan of any schedule derived with their labeling method is at most 22L+1 times larger than the makespan of an optimal schedule. This is a bit better than the tight ratio of 21L+1 achieved in this setting by arbitrary and “critical path”-oriented list schedules [3] to be described in Sect. 2.2. Moreover, the authors also argue that this superiority is preserved if the delays zero or L a task imposes may differ successor-wise. They further left as an open question whether the superiority even extends to the case where more general delays ij{0,,L} are allowed.

In this note, the suggestion mentioned first is however disconfirmed. To this end, an instance is given where, for any fixed L1, each task imposes only (both) the delays 0 and L on its successors, and such that the labeling by Bernstein, Rodeh and Gertner from [3] as well as the one used in the Coffman-Graham algorithm may lead to a solution whose quality approaches the bound of 21L+1 times the optimum. That is, as soon as delays are allowed to differ successor-wise, these labelings may be as bad as with an arbitrary list schedule which then also extends to delays ij{0,,L}. Moreover, due to the structure of our instance, the bound of 21L+1 also extends to some further list scheduling variants that refine their list order based on predecessor or successor counts.

Finally, the worst-case approximation ratio of labeling methods in the multiprocessor setting is investigated. In all conscience, it appears to be unknown whether the upper bound of 21m(L+1) proved in [21], [24] for arbitrary list scheduling on m2 processors can be improved for labeling methods. Also, a lower bound on the worst-case approximation guarantee of labeling methods appears to be unknown. Here, we provide an instance together with the corresponding analysis that induces a lower bound of 24m(L+1)+2 for m2.

The outline of this manuscript is as follows: In Sect. 2, we briefly classify general, level-based, and labeling-based list scheduling algorithms in the presence of delayed precedence constraints. We then give an overview of the prior and novel approximation results for these techniques in Sect. 3. Sect. 4 deals with a worst-case analysis of the introduced labeling methods proving the announced new results on their approximation guarantees. We close with a conclusion (Sect. 5).

Section snippets

Latencies, lists, levels and labels

In order to formally define the extensions of list scheduling and some of its specializations to the case of delayed precedence constraints, let us first denote an instance I of problem P|pj=1,prec(ij)|Cmax by the triple I(T,,). We will also employ the common directed acyclic graph (DAG) representation of an instance. In such a DAG G=(V,A,), the vertices V model the set of tasks T, and there is an arc (i,j)A with weight ij whenever ij. We will identify each task with its representing

Approximation guarantees for labeling methods

As already indicated in the introduction, there is some evidence that labeling methods can provide a superior worst-case performance compared to purely level-based or even arbitrary list schedules.

In the absence of delays (problem P|pj=1,prec|Cmax) for example, the labeling algorithm by Coffman and Graham [7] is optimal for m=2, and for m3 it has a worst-case approximation guarantee of 22m [4], [11], [20].2

Worst-case analysis

We first consider exactly the problem studied by Bernstein, Rodeh and Gertner in [3], i.e., a single processor and delays ij{0,L} for an arbitrary L1 – except that the delays are here defined arc-based instead of vertex-based. This precisely reflects the difference that the delays a task imposes on its successors may be non-uniform.

Theorem 4.1

If arc-wise different delays are permitted, the worst-case approximation ratio of a list schedule based on a BRG- or CG-labeling for problem 1|pj=1,prec(ij{0,L})

Conclusion

We answered in the negative the open question whether on a single processor the labeling proposed by Bernstein, Rodeh and Gertner [3] leads to superior list schedules compared to other techniques when confronted with arc-wise different precedence delays of zero or a fixed positive integer L. To the contrary, for problem 1|pj=1,prec(ij)|Cmax, and delays L1, the worst-case performance is bounded from below by 21L+1, i.e., no better than in the case of general list scheduling. Moreover, for P|pj

CRediT authorship contribution statement

Sven Mallach: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Validation, Visualization, Writing – original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Sven Mallach is a postdoctoral researcher at the University of Bonn in Germany. He received his PhD from the University of Cologne, Germany, in 2015. His research interests are in particular integer programming and combinatorial optimization. Sven Mallach‘s publications span across several applications in scheduling, sequencing and ordering, compiler optimization, maximum cut, graph theory and graph drawing, and linearization techniques for binary quadratic optimization problems. Currently, he

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  • Sven Mallach is a postdoctoral researcher at the University of Bonn in Germany. He received his PhD from the University of Cologne, Germany, in 2015. His research interests are in particular integer programming and combinatorial optimization. Sven Mallach‘s publications span across several applications in scheduling, sequencing and ordering, compiler optimization, maximum cut, graph theory and graph drawing, and linearization techniques for binary quadratic optimization problems. Currently, he is part of the High Performance Computing and Analytics Lab of the University of Bonn.

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