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Engineering Optimal Parallel Task Scheduling

Matthew Akram, Nikolai Maas, Peter Sanders, Dominik Schreiber

TL;DR

This work improves existing techniques for optimal P||C_max scheduling with a combination of new theoretical insights and careful practical engineering, and derives techniques to prune vast portions of the search space of branch-and-bound (BnB) approaches.

Abstract

The NP-hard scheduling problem P||C_max encompasses a set of tasks with known execution time which must be mapped to a set of identical machines such that the overall completion time is minimized. In this work, we improve existing techniques for optimal P||C_max scheduling with a combination of new theoretical insights and careful practical engineering. Most importantly, we derive techniques to prune vast portions of the search space of branch-and-bound (BnB) approaches. We also propose improved upper and lower bounding techniques which can be combined with any approach to P||C_max. Moreover, we present new benchmarks for P||C_max, based on diverse application data, which can shed light on aspects which prior synthetic instances fail to capture. In an extensive evaluation, we observe that our pruning techniques reduce the number of explored nodes by 90$\times$ and running times by 12$\times$. Compared to a state-of-the-art ILP-based approach, our approach is preferable for short running time limits and for instances with large makespans.

Engineering Optimal Parallel Task Scheduling

TL;DR

This work improves existing techniques for optimal P||C_max scheduling with a combination of new theoretical insights and careful practical engineering, and derives techniques to prune vast portions of the search space of branch-and-bound (BnB) approaches.

Abstract

The NP-hard scheduling problem P||C_max encompasses a set of tasks with known execution time which must be mapped to a set of identical machines such that the overall completion time is minimized. In this work, we improve existing techniques for optimal P||C_max scheduling with a combination of new theoretical insights and careful practical engineering. Most importantly, we derive techniques to prune vast portions of the search space of branch-and-bound (BnB) approaches. We also propose improved upper and lower bounding techniques which can be combined with any approach to P||C_max. Moreover, we present new benchmarks for P||C_max, based on diverse application data, which can shed light on aspects which prior synthetic instances fail to capture. In an extensive evaluation, we observe that our pruning techniques reduce the number of explored nodes by 90 and running times by 12. Compared to a state-of-the-art ILP-based approach, our approach is preferable for short running time limits and for instances with large makespans.
Paper Structure (24 sections, 3 theorems, 9 equations, 7 figures, 4 tables)

This paper contains 24 sections, 3 theorems, 9 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work.
  4. Pruning the Search Space of P||Cmax
  5. Prior Pruning Rules.
  6. New Pruning Rules.
  7. Efficient Computation.
  8. SAT-inspired Learning of Dead-ends.
  9. Bounding Techniques
  10. Lower Bounds.
  11. Lift++.
  12. Upper Bounds.
  13. LPT++.
  14. MS++.
  15. S$^4$.
  16. ...and 9 more sections

Key Result

Theorem 3.1

Consider the case that the FUR applies to job $j_i$ and processor $p_x$ and let $A' := A \cup \{ a_i = x \}$ be the partial assignment created by it. If an optimal completion of $A'$ has makespan $U'$ such that $U' > C_x^A + w_i$, then it is also an optimal completion of $A$.

Figures (7)

  • Figure 1: The two assignments on the left are equivalent when re-ordering the machines and then focusing on the silhouette. Numbers correspond to job indices.
  • Figure 2: BnB algorithm. Diamonds represent "if-then-else" nodes; green filled arrows represent true-branches, orange non-filled arrows represent false-branches. Where applicable, diamonds are labelled with the rule or the theorem enabled by the respective condition. Global fields common to all recursive calls are $U$ and $L$, irrelevance index $\rho$, the RET, state table ST, and best solution $S$.
  • Figure 3: Results of bounding experiments. Red denotes upper bounds, blue denotes lower bounds. Left: Bounding quality. Each curve represents the quotients between a technique's found bound and the corresponding optimal makespan, sorted by their difference to 1 and normalised to the $x$ interval $[0,1]$. Right: Cumulative running time distributions of all bounding techniques (higher is better).
  • Figure 4: Cumulative running times for BnB and ILP approaches, disregarding instances solved exactly by bounding. Note the log. scale in $x$ direction. VBS ("Virtual Best Solver") depicts a virtual running time oracle combining CDSM and ILP.
  • Figure 5: Direct comparison of running times of Base vs. FUR (left), FUR vs. CDSM (center), and CDSM vs. the ILP approach (right). Note the logarithmic scale in both directions.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 2.1