From Estimates to Schedules: Learning-Augmented Restricted Assignment

Abstract: In this work, we study Restricted Assignment scheduling on multiple machines, where each job can be processed only on a specified subset of machines and the objective is to minimize the makespan. We introduce a learning-augmented setting in which a possibly infeasible predicted assignment is provided. The prediction error (moved-load) is measured by the total processing volume that must be reassigned in order to obtain an optimal feasible schedule. Using a single prediction, we obtain two types of guarantees. First, we design an algorithm whose approximation ratio degrades smoothly with the prediction error while retaining a worst-case guarantee independent of the prediction quality. More precisely, for any fixed constant, we can make the additive dependence on the prediction error arbitrarily small, at the cost of increasing the polynomial running time. This guarantee can also be combined with any approximation algorithm for the problem without predictions to obtain robustness. Second, given a makespan estimate, we provide a repair procedure that returns a schedule matching this estimate in time parameterized by the prediction error. This allows the algorithm to exploit the separation between estimation and approximation algorithms for Restricted Assignment. Finally, we complement the repair algorithm with a parameterized hardness result, showing that exact moved-load repair with a given target makespan is W[1]-hard when parameterized by the amount of moved-load.

• The paper presents a framework that integrates possibly infeasible predictions into Restricted Assignment scheduling to achieve makespan guarantees that degrade smoothly with prediction error.
• It introduces innovative error metrics (moved-load and moved-job) that quantify the divergence from optimal assignments, enabling effective error-dependent repair and LP rounding techniques.
• The robustness is ensured by combining a learning-augmented method with an α-approximation algorithm, guaranteeing performance no worse than traditional worst-case bounds.

Learning-Augmented Approaches to Restricted Assignment Scheduling

Problem Setup and Motivation

The paper "From Estimates to Schedules: Learning-Augmented Restricted Assignment" (2606.08377) studies the offline Restricted Assignment scheduling problem: each job has a processing time and a subset of machines to which it can be assigned, and the objective is to minimize the makespan. The classical setting is notoriously intractable, with no polynomial-time approximation better than $3/2$ unless $\mathrm{P}=\mathrm{NP}$, and the best known approximation ratio remaining at $2$ [LST90]. Motivated by advances in leveraging predictions in algorithm design, the paper investigates integrating (possibly infeasible) predicted assignments into the Restricted Assignment problem via learning-augmented algorithms, with the goal of achieving improved performance when predictions are accurate while preserving strong worst-case guarantees.

Prediction Model and Error Measurement

The key aspect of the framework is the introduction of a moved-load error metric, defined as the minimal total processing time that must be reassigned from a predicted assignment to transform it into an optimal feasible schedule. This error measure allows tracking the divergence between the predicted solution and the true optimum without requiring prediction feasibility. The framework also defines an analogous moved-job error metric, counting the number of jobs that must be reassigned, and supports both load- and job-based error measures.

Smooth and Robust Approximation Guarantees

Given a (possibly infeasible) predicted assignment $\hat\pi$, the main result is the design of scheduling algorithms whose makespan guarantees degrade smoothly with the prediction error $E$. Concretely, for any $\delta \in (0,1]$, the algorithm outputs a feasible schedule with makespan at most $OPT + \delta E$ in time $(nm)^{O(1/\delta)}$, where $n$ is the number of jobs and $m$ is the number of machines. The algorithm constructs this guarantee by (i) projecting the predicted assignment into feasibility and (ii) exhaustively guessing a small set of large jobs on which the prediction significantly diverges from optimum, combining this with LP rounding for small jobs.

Crucially, the method can be robustified: by running an $\mathrm{P}=\mathrm{NP}$0-approximation algorithm for Restricted Assignment in parallel and returning the best solution, the makespan is always at most $\mathrm{P}=\mathrm{NP}$1. The robustness ensures that the algorithm never performs worse than the classical approximation ratio, regardless of prediction quality.

Error-Dependent Repair and Parameterized Algorithms

The paper develops an error-parameterized repair algorithm: given a makespan estimate $\mathrm{P}=\mathrm{NP}$2, the procedure either certifies the infeasibility of such a schedule or returns a solution with makespan at most $\mathrm{P}=\mathrm{NP}$3, with a running time that depends exponentially on $\mathrm{P}=\mathrm{NP}$4, the minimum moved-load required to achieve makespan $\mathrm{P}=\mathrm{NP}$5 from the predicted assignment. Specifically, the algorithm achieves running time $\mathrm{P}=\mathrm{NP}$6, significantly improving over naive $\mathrm{P}=\mathrm{NP}$7 enumeration.

This repair approach exploits the fact that, for Restricted Assignment, the best known estimation (finding a value within $\mathrm{P}=\mathrm{NP}$8 of the optimum) and approximation ratios differ: estimation achieves $\mathrm{P}=\mathrm{NP}$9 [lars_2], while no approximation algorithm betters $2$0. Thus, if an external estimation subroutine or domain knowledge supplies a makespan estimate $2$1 with $2$2, the repair procedure can recover provably better schedules in error-dependent time.

The repair procedures are supported by dynamic programs that reduce to multi-dimensional knapsack over small "incident" sets of machines, determined by the jobs whose assignment must be changed. The analysis exploits structural properties of moved-load and moved-job errors, yielding fine-grained control over the algorithm's state space.

Parameterized Hardness

The scope of error-parameterized repair is shown to be tight: the authors prove that exact repair with a given moved-load or moved-job budget (for a target makespan) is W[1]-hard parameterized by the amount of moved-load. This implies, under standard complexity assumptions, that no fixed-parameter tractable ($2$3) algorithm is possible in $2$4 alone, unless $2$5. The reduction is robust, even for feasible predictions and instances with at most three distinct processing times.

Instantiations and Extensions

Applying the general framework to Restricted Assignment and its two-valued variant (jobs with one of two processing times), the results enable practitioners to interpolate between the best possible estimation and approximation ratios based on the prediction error. Specifically:

• For Restricted Assignment, the robust guarantee yields makespan at most $2$6, and error-dependent repair can match the $2$7 estimation ratio in time exponential in $2$8.
• For two-valued Restricted Assignment, the same paradigm yields a schedule of makespan $2$9, or, using repair, matches the polynomial-time estimation ratio ($\hat\pi$0) for small error.

Further, the repair framework is generalized to moved-job errors, enabling more favorable running times in instances where only a small number of large jobs are misplaced by the predictor.

Implications and Future Directions

The techniques establish a new synthesis of learning-augmented and parameterized approximation for NP-hard scheduling problems with side information. The methods provide a spectrum of algorithms interpolating between standard worst-case approximation and optimal solutions when predictions are sufficiently accurate and the error is small. Notably, the framework cleanly separates the estimation and repair stages, which is particularly beneficial in domains with recurring structure or historical data likely to yield strong predictions.

The authors also highlight extensions to allocation problems with max-min objectives, e.g., the Restricted Assignment Santa Claus problem, where the estimation/approximation ratio separation is even more pronounced and the structural machinery developed may yield similarly strong guarantees.

Conclusion

The paper presents a comprehensive framework for learning-augmented scheduling in the Restricted Assignment setting, integrating both smoothness and robustness to prediction errors. By introducing principled error measures and coupling them to advanced algorithmic techniques, the authors derive algorithms that achieve strong performance guarantees in a regime interpolating between traditional approximation and exact optimization, parameterized by the quality of prediction. The tight parameterized hardness results delineate the boundary of tractability, while the general repair paradigm points the way toward further advances in learning-augmented combinatorial optimization for complex scheduling and allocation problems.