A Simpler QPTAS for Scheduling Jobs with Precedence Constraints

2025-01-15

Embedded Systems and Applications

AI Summary

This paper studies the NP-hard problem of minimizing makespan for unit-length jobs with precedence constraints on three identical parallel machines. For this classical scheduling problem, we present the first concise quasi-polynomial-time approximation scheme (QPTAS) that avoids both LP hierarchies and dyadic partitioning. Our approach directly guesses the relative positions of critical jobs in an optimal schedule, then applies interval-based recursive partitioning coupled with a greedy assignment strategy to achieve a $(1+varepsilon)$-approximation. The algorithm runs in time $n^{O_{varepsilon}(log^3 log n)}$. Compared to prior QPTASes, our method significantly improves interpretability and analytical simplicity, offering the most intuitive and transparent quasi-polynomial-time approximation scheme known to date for this problem.

Technology Category

• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study the classical scheduling problem of minimizing the makespan of a set of unit size jobs with precedence constraints on parallel identical machines. Research on the problem dates back to the landmark paper by Graham from 1966 who showed that the simple List Scheduling algorithm is a $(2-frac{1}{m})$-approximation. Interestingly, it is open whether the problem is NP-hard if $m=3$ which is one of the few remaining open problems in the seminal book by Garey and Johnson. Recently, quite some progress has been made for the setting that $m$ is a constant. In a break-through paper, Levey and Rothvoss presented a $(1+epsilon)$-approximation with a running time of $n^{(log n)^{O((m^{2}/epsilon^{2})loglog n)}}$[STOC 2016, SICOMP 2019] and this running time was improved to quasi-polynomial by Garg[ICALP 2018] and to even $n^{O_{m,epsilon}(log^{3}log n)}$ by Li[SODA 2021]. These results use techniques like LP-hierarchies, conditioning on certain well-selected jobs, and abstractions like (partial) dyadic systems and virtually valid schedules. In this paper, we present a QPTAS for the problem which is arguably simpler than the previous algorithms. We just guess the positions of certain jobs in the optimal solution, recurse on a set of guessed subintervals, and fill in the remaining jobs with greedy routines. We believe that also our analysis is more accessible, in particular since we do not use (LP-)hierarchies or abstractions of the problem like the ones above, but we guess properties of the optimal solution directly.

Problem

• Scheduling
• NP-hardness
• Parallel Computing

Innovation

• Simplified Scheduling Method
• Intuitive Rule-based Approach
• Recursive Subproblem Solving