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.

翻译：在本文中，我们研究了多机上的受限分配调度问题，其中每个作业只能在指定的机器子集上处理，目标是最大化最小化完工时间。我们引入了一个学习增强设置，其中提供了可能不可行的预测分配。预测误差（移动负载）通过为获得最优可行调度而必须重新分配的总体处理量来度量。利用单一预测，我们获得了两类保证。首先，我们设计了一种算法，其近似比随预测误差平滑退化，同时保留与预测质量无关的最坏情况保证。更精确地说，对于任意固定常数，我们可以使对预测误差的附加依赖任意小，代价是增加多项式运行时间。该保证还可以与任何无预测问题的近似算法结合以实现鲁棒性。其次，给定一个完工时间估计，我们提供了一种修复程序，该程序在由预测误差参数化的时间内返回与此估计匹配的调度。这使得算法能够利用受限分配中估计算法与近似算法之间的分离。最后，我们用参数化难度结果补充了修复算法，表明当以移动负载量为参数时，在给定目标完工时间下实现精确移动负载修复是W[1]-难的。