A simpler QPTAS for scheduling jobs with precedence constraints
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916567538728960 |
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| author | Das, Syamantak Wiese, Andreas |
| author_facet | Das, Syamantak Wiese, Andreas |
| contents | 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 $(1+ε)$-approximation with a running time of $n^{(\log n)^{O((m^{2}/ε^{2})\log\log n)}}$[STOC
2016, SICOMP 2019] and this running time was improved to quasi-polynomial
by Garg[ICALP 2018] and to even $n^{O_{m,ε}(\log^{3}\log n)}$
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09091 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A simpler QPTAS for scheduling jobs with precedence constraints Das, Syamantak Wiese, Andreas Data Structures and Algorithms 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 $(1+ε)$-approximation with a running time of $n^{(\log n)^{O((m^{2}/ε^{2})\log\log n)}}$[STOC 2016, SICOMP 2019] and this running time was improved to quasi-polynomial by Garg[ICALP 2018] and to even $n^{O_{m,ε}(\log^{3}\log n)}$ 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. |
| title | A simpler QPTAS for scheduling jobs with precedence constraints |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2501.09091 |