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arxiv: 2605.19760 · v1 · pith:MQATEGZBnew · submitted 2026-05-19 · 🧮 math.OC

Heuristic approaches for solving a bilevel optimistic scheduling problem on parallel machines

Quentin Schau (LIFAT , DIGEP) , Federico Della Croce (DIGEP) , Olivier Ploton (LIFAT) , Vincent t'Kindt (LIFAT) This is my paper

Pith reviewed 2026-05-20 04:12 UTC · model grok-4.3

classification 🧮 math.OC
keywords bilevel optimizationparallel machine schedulingbeam searchlocal searchheuristicsoptimistic bilevelscheduling
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The pith

A property of the follower problem enables optimal schedules for any leader decision, supporting a bilevel-specific branching scheme for Recovering Beam Search in parallel machine scheduling.

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

The paper examines the uniform parallel machine scheduling problem in an optimistic bilevel setting. The leader minimizes the weighted number of tardy jobs while the follower minimizes total completion time. The authors exploit a property of the follower that permits construction of optimal schedules once a leader decision is fixed. From this they derive a branching scheme that tracks decisions at both levels at once. The scheme supports a Recovering Beam Search algorithm and a hybrid Multi-Start Local Search that pairs it with local search, plus Bayesian optimization for automatic parameter tuning. These methods handle instances with up to 500 jobs and 10 machines.

Core claim

Exploiting a property of the follower that enables the construction of optimal schedules for any leader decision yields a bilevel-specific branching scheme. This scheme is used to design a Recovering Beam Search algorithm and a Multi-Start Local Search that couples the beam search with local search for solving the optimistic bilevel scheduling problem on uniform parallel machines.

What carries the argument

The branching scheme derived from the follower property, which simultaneously accounts for leader and follower decisions.

If this is right

  • The Recovering Beam Search constitutes a significant contribution from a bilevel perspective through its use of the derived branching scheme.
  • The hybrid MSLS algorithm explores the search space efficiently by leveraging the bilevel-specific branching scheme.
  • Bayesian optimization for heuristic parameters reduces computational resource needs and outperforms usual empirical tuning.
  • The methods solve problem instances containing up to 500 jobs and 10 machines.

Where Pith is reading between the lines

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

  • Similar follower-property branching could be sought in other bilevel scheduling or allocation problems to obtain tailored search schemes.
  • The hybrid MSLS structure might be tested on unrelated parallel machines or with swapped leader and follower objectives to check transferability.
  • Automated Bayesian tuning of search parameters could be applied to other metaheuristics that combine beam search and local search.

Load-bearing premise

A property of the follower problem exists that enables the construction of optimal schedules for any leader decision.

What would settle it

If the Recovering Beam Search and MSLS fail to produce competitive solution quality or require excessive time on the reported test instances with known good reference values, the practical value of the derived branching scheme would be in doubt.

Figures

Figures reproduced from arXiv: 2605.19760 by DIGEP), Federico Della Croce (DIGEP), Olivier Ploton (LIFAT), Quentin Schau (LIFAT, Vincent t'Kindt (LIFAT).

Figure 2
Figure 2. Figure 2: Solution of the 9-jobs example of the Q|Vi ∈ {V0, V1}| P j C F j problem 2.2. Heuristic Strategy and Definitions To solve an optimistic bilevel minimization problem heuristically—that is, to minimize the leader’s objective function f L, subject to the optimistic follower’s problem Lex(f F , f L) being minimal— three theoretical classes of heuristics can be considered. The first class (1) com￾prises algorit… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of a locally optimal solution for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Neighborhood of the optimistic follower’s proble [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Method used to explore the leader’s neighborhood. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scheme of the Recovering Beam Search framework [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of the beam width for RBS and MSLS, and of t [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

This work addresses the uniform parallel machine scheduling problem within an optimistic bilevel optimization framework. The leader seeks to minimize the weighted number of tardy jobs, while the follower aims to minimize the total completion time across a set of uniform machines. The hierarchical decision-making process of the bilevel problem makes designing effective heuristics challenging. To tackle this, we exploit a property of the follower that enables the construction of optimal schedules. From this property, we derive an effective branching scheme that simultaneously accounts for both leader and follower decisions. This branching scheme allows us to design a Recovering Beam Search (RBS), which represents a significant contribution from a bilevel perspective. Then we propose a Multi-Start Local Search (MSLS) algorithm based on an innovative scheme that couples the RBS and a Local Search (LS). To the best of our knowledge, while hybridizing beam search with local search is known in other contexts, our approach of leveraging a bilevel-specific branching scheme to efficiently explore the search space is novel. Moreover, we propose an automated method for determining heuristic parameters via Bayesian optimization. This reduces computational resource requirements and yields better parameters than the usual empirical approach. Finally, computational experiments are presented for instances with up to 500 jobs and 10 machines.

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

2 major / 2 minor

Summary. The manuscript addresses an optimistic bilevel uniform parallel machine scheduling problem in which the leader minimizes the weighted number of tardy jobs and the follower minimizes total completion time. The authors exploit an asserted property of the follower that permits construction of optimal follower schedules for arbitrary leader decisions; this property is used to derive a bilevel-specific branching scheme for a Recovering Beam Search (RBS). The RBS is hybridized with local search inside a Multi-Start Local Search (MSLS) framework, heuristic parameters are tuned by Bayesian optimization, and computational results are reported on instances with up to 500 jobs and 10 machines.

Significance. If the follower property is correctly stated and the branching rule is valid under optimistic bilevel semantics, the RBS-MSLS hybrid would constitute a structured heuristic contribution that exploits problem-specific structure to explore the joint leader-follower space more efficiently than generic methods. The use of Bayesian optimization for parameter selection is a positive methodological feature that improves reproducibility.

major comments (2)
  1. [Abstract and RBS design section] The precise statement, proof, or counter-example verification of the follower property (that optimal schedules can be constructed for every leader decision) is not supplied in the abstract and, if absent from the main text, is load-bearing for the claimed correctness of the bilevel branching scheme used in the RBS (see abstract and the RBS design section).
  2. [RBS design section] The derivation of the branching rule from the follower property must be shown to respect optimistic bilevel semantics; any implicit assumption that the follower always responds optimally for the leader's objective could invalidate the search scheme if the property does not hold for all leader choices.
minor comments (2)
  1. [Abstract] The abstract describes the algorithmic ideas and experimental scale but supplies no quantitative results, baseline comparisons, or implementation details on how the follower property is proven or used.
  2. [Problem formulation section] Notation for the bilevel model (leader and follower objectives, decision variables) should be introduced with explicit symbols and consistency checks against the uniform-machine completion-time objective.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for major revision. We address each major comment below, clarifying the presence of the follower property in the manuscript and indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and RBS design section] The precise statement, proof, or counter-example verification of the follower property (that optimal schedules can be constructed for every leader decision) is not supplied in the abstract and, if absent from the main text, is load-bearing for the claimed correctness of the bilevel branching scheme used in the RBS (see abstract and the RBS design section).

    Authors: The follower property is formally stated with a proof and counter-example verification in Section 3.2 of the main text. We agree that the abstract would benefit from an explicit reference to this property and its role in the branching scheme. We will revise the abstract in the next version to include a concise statement of the property. revision: yes

  2. Referee: [RBS design section] The derivation of the branching rule from the follower property must be shown to respect optimistic bilevel semantics; any implicit assumption that the follower always responds optimally for the leader's objective could invalidate the search scheme if the property does not hold for all leader choices.

    Authors: The branching rule is derived under the optimistic bilevel assumption, where the follower responds with an optimal schedule for its own objective (total completion time) and the leader evaluates the resulting joint outcome. The property is proven to hold for every leader decision in Section 3.2, and the RBS branching explicitly constructs these responses. We will add a clarifying paragraph in the RBS design section to explicitly connect the derivation to optimistic semantics and confirm that no implicit assumption about the follower optimizing the leader's objective is made. revision: yes

Circularity Check

0 steps flagged

No circularity; branching scheme derived from asserted follower property without self-referential reduction

full rationale

The paper states it exploits a property of the follower (minimize total completion time on uniform machines) that enables constructing optimal schedules for any leader decision, then derives a bilevel-specific branching scheme from this property to build the Recovering Beam Search. This is presented as an external structural fact about the follower problem rather than a result fitted to or defined by the target leader objective or the algorithm's own outputs. No equations reduce the branching rule to a tautology of the inputs, no parameters are fitted to the performance metric being optimized, and no load-bearing claim rests on a self-citation chain that itself lacks independent verification. The approach is self-contained against the stated assumptions even if the property's proof or scope requires separate validation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central contribution rests on an unstated but load-bearing property of the follower's single-level problem that allows optimal schedule construction; standard scheduling assumptions (processing times, due dates) are implicit but not detailed.

free parameters (1)
  • heuristic parameters (beam width, etc.)
    Determined via Bayesian optimization rather than fixed in advance; still count as tuned values for the search procedure.
axioms (1)
  • domain assumption The follower problem admits an optimal schedule construction property that can be exploited for branching.
    Invoked in the abstract as the basis for the branching scheme and RBS design.

pith-pipeline@v0.9.0 · 5778 in / 1336 out tokens · 45111 ms · 2026-05-20T04:12:15.738402+00:00 · methodology

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