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arxiv: 2607.00878 · v1 · pith:P2K7FVQPnew · submitted 2026-07-01 · 💻 cs.DS

Improved Approximation Algorithms for Parallel Task Scheduling and Multiple Cluster Scheduling

Bennet Edler , Klaus Jansen , Felix Ohnesorge , Lis Pirotton This is my paper

Pith reviewed 2026-07-02 04:20 UTC · model grok-4.3

classification 💻 cs.DS
keywords approximation algorithmsparallel task schedulingmultiple cluster schedulingmakespan minimizationscheduling theoryNP-hard problems
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The pith

A new O(n log n) algorithm achieves (4/3)OPT + p_max for parallel task scheduling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives an algorithm for Parallel Task Scheduling that produces schedules with makespan at most (4/3)OPT + p_max and runs in O(n log n) time. This directly answers the open question whether the previous (3/2)OPT + p_max bound from Jansen and Rau could be improved. The same work speeds up the known 2-approximation for the Multiple Cluster Scheduling extension and removes the divisibility restriction on the number of clusters for the 9/4 approximation. The 2-approximation is shown to be tight because no better ratio exists unless P equals NP. An implementation is also provided to demonstrate that the methods work on concrete instances.

Core claim

We present an algorithm for Parallel Task Scheduling that achieves an approximation ratio of (4/3)OPT + p_max with running time O(n log n). For Multiple Cluster Scheduling we improve the running time of the 2-approximation and generalize the 9/4 approximation to arbitrary numbers of clusters. The 2-approximation for MCS is tight since one cannot hope for an approximation ratio better than 2 unless P=NP.

What carries the argument

A near-linear-time approximation algorithm that schedules jobs with given processing times and machine requirements by efficient partitioning and assignment to minimize makespan.

If this is right

  • The (4/3)OPT + p_max guarantee for PTS is now available in near-linear time.
  • The 9/4 approximation for MCS holds for every number of clusters rather than only when N is divisible by 3.
  • The 2-approximation for MCS cannot be improved by any polynomial-time algorithm unless P=NP.
  • The implemented algorithms can be used directly on practical scheduling instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same partitioning technique may adapt to related resource-constrained scheduling problems such as moldable tasks or heterogeneous clusters.
  • Tighter analysis of the additive p_max term could remove or reduce the additive term in future work.
  • Empirical testing on very large instances would show whether the observed running time stays close to the O(n log n) bound in practice.

Load-bearing premise

The standard parallel-task model admits an algorithm whose analysis yields exactly the stated (4/3)OPT + p_max bound without hidden restrictions on job sizes or machine counts.

What would settle it

An explicit list of jobs with processing times and machine requirements where every feasible schedule has makespan strictly larger than (4/3)OPT + p_max.

Figures

Figures reproduced from arXiv: 2607.00878 by Bennet Edler, Felix Ohnesorge, Klaus Jansen, Lis Pirotton.

Figure 1
Figure 1. Figure 1: Constructed schedules with sparse intervals. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schedule 𝜎1 Firstly, there is one important observation, regarding the place￾ment of small and medium jobs. By construction, the machine requirement of those jobs is non-decreasing in time. This implies that the placement of a job 𝑗 is not affected by the presence of any job 𝑘 with 𝑞𝑘 < 𝑞𝑗 this implies: 𝑑𝑜𝑚(𝜎1 (J \ S)) = 𝑑𝑜𝑚(𝜎1 (J )) \ S. (1) In addition, note that 𝐼1 might be of arbitrary height. Now, we … view at source ↗
Figure 3
Figure 3. Figure 3: Schedule 𝜎1 with partitioning into 𝜎 𝐵 1 and 𝜎 𝑇 1 To prove Lemma 3.4, we iteratively apply Claims 3.5 and 3.7 until |𝐼1 | < 𝑝max. We first note that each application of these claims removes at least one job. Consequently, the procedure is guaranteed to terminate. We have shown that by reintroducing all the jobs removed via Claim 3.5, we do not increase the approximation ratio. Then, reinserting all jobs r… view at source ↗
Figure 4
Figure 4. Figure 4: Schedule 𝜎2 after Algorithms 4 and 5 Algorithm 4 𝜎 ′ 2 : Scheduling the Remaining Jobs in M 1: Place all jobs 𝑗 ∈ M via GreedyScheduling on two stacks. Lemma 3.10. The height of the schedules 𝜎1 and 𝜎 ′ 2 , constructed by Algorithms 3 and 4, satisfies: ℎ(𝜎1) + ℎ(𝜎 ′ 2 ) ≤ OPT + |𝐼1 | 2 + |𝐼 ′ 2 | 2 Proof. For simplicity, we start with the assumption S = ∅. Consider now the schedule 𝜎1 and the intermediate … view at source ↗
Figure 5
Figure 5. Figure 5: Illustrations around Algorithm 6 Lemma 3.13. Given the schedules 𝜎 𝐵 1 , 𝜎 𝑇 1 , and 𝜎2, as constructed by Algorithm 6, we can construct a schedule 𝜎 with ℎ(𝜎) ≤ 4 3 OPT+𝑝max. Proof. To build 𝜎, we first alter 𝜎2 by removing all small jobs which complete after the last tiny job has completed and remove them. Then, we reschedule those jobs by GreedyScheduling on top. Let 𝑗 be the tiny job with the highest c… view at source ↗
Figure 6
Figure 6. Figure 6: 𝜎1 and 𝜎2 (mirrored) as in Observation 3.14 (or Observation 3.11). Or the height does change, then we can bound the area deficit to 3 4 𝑝max. Construct the new schedule 𝜎 by placing 𝜎1 at 0 and (rotated) 𝜎2 at ℎ(𝜎1) − ℎ ′ such that ℎ ′ ∈ Q is maximal. The resulting schedule is illustrated in Figure 6c. Note that in 𝜎 the height of both sparse intervals 𝐼1 and 𝐼2 is reduced by exactly ℎ ′ . Denote now the h… view at source ↗
Figure 7
Figure 7. Figure 7: Testing the Parallel Task Scheduling algorithm. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

In the problem of Parallel Task Scheduling (PTS), we are asked to schedule $n$ jobs, each with a fixed processing time and machine requirement, such that the completion time of the last job is minimized. Jansen and Rau (2019) presented an algorithm for PTS that achieves an approximation ratio of $(3/2)\text{OPT} + p_{\max}$. They additionally posed the open question whether an approximation ratio of $(4/3)\text{OPT} + p_{\max}$ is possible. In this work, we present such an algorithm with a running time of $O(n\log n)$. The problem of Multiple Cluster Scheduling (MCS) is a natural extension of PTS where we are given $N$ clusters each of $m$ machines to schedule jobs. Jansen and Rau (2019) adapted their PTS algorithm to MCS with the following results: (1) a 2 approximation, and (2) a near-linear 9/4 approximation if $N$ is divisible by 3. We improve the running time of their 2-approximation and generalize the 9/4 approximation to the general case. The 2-approximation for MCS is tight, since one cannot hope for an approximation ratio better than 2, unless P=NP [Zhuk, 2006]. In addition to our theoretical results, we implement our algorithm and show its practical applicability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims an O(n log n)-time algorithm for Parallel Task Scheduling (PTS) achieving approximation ratio (4/3)OPT + p_max, resolving an open question of Jansen and Rau (2019). It also improves the running time of the 2-approximation for Multiple Cluster Scheduling (MCS), generalizes the 9/4-approximation to arbitrary N, notes that 2 is tight for MCS unless P=NP, and reports an implementation demonstrating practical applicability.

Significance. If the claims hold, the work would resolve a stated open question with a near-linear-time algorithm and a strictly improved guarantee for the standard PTS model (fixed p_j, m_j, makespan objective). The explicit credit to the prior (3/2)OPT + p_max result and the acknowledgment that the MCS 2-approximation is tight are appropriate. The inclusion of an implementation is a positive feature for a theory paper in this area.

major comments (1)
  1. [Abstract] Abstract: the central claims—an O(n log n) algorithm with exactly (4/3)OPT + p_max for PTS and the stated MCS improvements—are asserted without any algorithm description, pseudocode, proof sketch, reduction, or experimental data. This prevents verification of the claimed guarantee or running time and is load-bearing for the paper's main contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. Below we respond point-by-point to the single major comment.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims—an O(n log n) algorithm with exactly (4/3)OPT + p_max for PTS and the stated MCS improvements—are asserted without any algorithm description, pseudocode, proof sketch, reduction, or experimental data. This prevents verification of the claimed guarantee or running time and is load-bearing for the paper's main contribution.

    Authors: Abstracts are concise summaries by design and are not intended to contain algorithm descriptions, pseudocode, proof sketches, reductions, or experimental data; those elements appear in the body of the manuscript (algorithm, analysis, implementation, and experimental results). The abstract accurately summarizes the results established in the full paper without overclaiming. revision: no

Circularity Check

0 steps flagged

No significant circularity in algorithmic construction

full rationale

The paper presents an independent algorithmic construction achieving the (4/3)OPT + p_max guarantee in O(n log n) time for the standard PTS model, improving on the prior (3/2)OPT + p_max result from Jansen and Rau (2019). The central claim does not reduce to any fitted parameter, self-definition, or load-bearing self-citation; the 2-approximation tightness for MCS is supported by external citation (Zhuk 2006). No equations or reductions loop back to the paper's own inputs, and the result is presented as a new construction without ansatz smuggling or renaming of known patterns. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities appear in the abstract; the contribution is a claimed algorithmic construction within the standard scheduling model.

pith-pipeline@v0.9.1-grok · 5792 in / 1159 out tokens · 30798 ms · 2026-07-02T04:20:06.847651+00:00 · methodology

discussion (0)

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Reference graph

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