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Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling

Sven Jäger, Guillaume Sagnol, Daniel Schmidt genannt Waldschmidt, Philipp Warode

TL;DR

This paper studies non‑clairvoyant scheduling for minimizing the sum of weighted completion times on a single machine, focusing on kill‑and‑restart and preemptive strategies. It introduces the $b$‑scaling strategy and its randomized variant, deriving tight competitive ratios via spectral analysis of Toeplitz matrices; a deterministic bound of $1+\dfrac{2b^{3/2}}{b-1}$ is achieved with a minimum of $1+3\sqrt{3}$ at $b=3$, and a randomized bound of $\dfrac{2b+\sqrt{b}-1}{\sqrt{b}\ln b}$ is minimized near $b\approx 8.16$, giving about $3.032$. The paper also proves a lower bound of $3-\dfrac{2}{n+1}$ for deterministic strategies, and demonstrates a $2$-competitive online rule (WSETF) for release‑date settings, with extensions to online release dates and parallel machines yielding sub‑10 guarantees. Together, these results show that constant‑competitive non‑clairvoyant strategies are achievable and provide a methodological framework using Toeplitz/eigenvalue analyses for future generalizations.

Abstract

We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of~$3$ for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any $b > 1$ a tight analysis for the natural $b$-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of $(1+3\sqrt{3})\approx 6.197$ for the deterministic and of $\approx 3.032$ for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is $2$-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than $10$ for adaptions of the $b$-scaling strategy to online release dates and unweighted jobs on identical parallel machines.

Competitive Kill-and-Restart and Preemptive Strategies for Non-Clairvoyant Scheduling

TL;DR

This paper studies non‑clairvoyant scheduling for minimizing the sum of weighted completion times on a single machine, focusing on kill‑and‑restart and preemptive strategies. It introduces the ‑scaling strategy and its randomized variant, deriving tight competitive ratios via spectral analysis of Toeplitz matrices; a deterministic bound of is achieved with a minimum of at , and a randomized bound of is minimized near , giving about . The paper also proves a lower bound of for deterministic strategies, and demonstrates a -competitive online rule (WSETF) for release‑date settings, with extensions to online release dates and parallel machines yielding sub‑10 guarantees. Together, these results show that constant‑competitive non‑clairvoyant strategies are achievable and provide a methodological framework using Toeplitz/eigenvalue analyses for future generalizations.

Abstract

We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of~ for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any a tight analysis for the natural -scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of for the deterministic and of for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is -competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than for adaptions of the -scaling strategy to online release dates and unweighted jobs on identical parallel machines.
Paper Structure (20 sections, 36 theorems, 144 equations, 3 figures, 1 algorithm)

This paper contains 20 sections, 36 theorems, 144 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Related Work.
  4. Non-preemptive scheduling.
  5. Preemptive scheduling.
  6. Kill-and-restart scheduling.
  7. Further related work.
  8. Preliminaries
  9. Kill-and-Restart Strategies.
  10. Lower Bound for Deterministic Strategies
  11. The b-Scaling Strategy
  12. Tight Analysis of the Deterministic b-Scaling Strategy
  13. Tight Analysis of the Randomized b-Scaling Strategy
  14. Weighted Shortest Elapsed Time First
  15. Extensions of the b-Scaling Strategy for Release Dates and Parallel Machines
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $1 \, \vert {} \vert \, \sum C_j$, no deterministic non-clairvoyant kill-and-restart strategy can achieve a competitive ratio smaller than $3-\frac{2}{n+1}$ on instances with $n\geq 3$ jobs, even if every job $j$ has processing time $p_j \ge 1$.

Figures (3)

  • Figure 1: An example for the $\mathrm{WSETF}$ schedule and the $\mathrm{PWSPT}$ schedule for the instance $I$ with $\boldsymbol{p} = (4,4,8,16,32,32)^\top$, $\boldsymbol{r} = (30,93,24,0,0,24)^\top$, and unit weights $\boldsymbol{w} = 1$. Thick lines indicate completions of jobs.
  • Figure 2: An example for the three schedules considered in the proof of \ref{['thm:trivial_bound_release_dates']} for the instance $I$ with $\boldsymbol{p} = (8,9,9,3,2,2)^{\top}$, $\boldsymbol{r} = (0,0,18,18,30,93)^{\top}$, and unit weights $\boldsymbol{w} = \boldsymbol{1}$, for $b = 2$. Gray areas indicate infinitesimal probing; thick lines indicate the completion of a job. Top: The schedule of $\mathfrak{D}_2 (I)$ for the original instance $I$. Middle: The schedule of $\mathfrak{D}_2 (I')$ for the modified instance $I'$ with processing times rounded to the next integer power of $b=2$. The release dates $r_3'$ and $r_4'$ are shifted such that they coincide with the end of a probing. Bottom: The schedule of $\mathrm{WSETF}$ for the instance $I"$. The processing times (corresponding to the colored areas) correspond to the total probing times in the schedule of $\mathfrak{D}_2 (I')$. The completion times in this schedule are higher or equal to the completion times in the schedule of $\mathfrak{D}_2 (I')$, as indicated by the vertical dashed lines.
  • Figure 3: Illustration of the situation in the proof of \ref{['lem:release_dates_I\'_I"']}. The release dates $r^{(i)}$ and the end points of the rounds in $\mathfrak{D}_{b}$ subdivide the time axis into intervals. The shaded intervals do not contain any successful probings. The value $q_{\min} = 5$ is chosen such that the end of the round $e^{(i)}(q_{\min} - 1)$ is before the next release date and such that between $r^{(i)}$ and $e^{(i)}(q_{\min} - 1)$ no job completes. \ref{['clm:qmin']} refers to the endpoints $e^{(i)}(q_{\min} - 1)$, while \ref{['clm:YWSETF-YALG']} refers to all subsequent endpoints of rounds. Note, that some endpoints (e.g, $e^{(2)}(6)$ or $e^{(2)} (7)$) lie after one or more subsequent release dates.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Proposition 2.1
  • proof
  • ...and 58 more