A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty

2026-02-02

AI Summary

This study investigates the parameterized complexity of minimizing the number of late jobs in single-machine scheduling under budgeted uncertainty, where the objective is to minimize the worst-case number of tardy jobs. Leveraging parameterized complexity theory, the authors combine pseudopolynomial algorithms with reductions from the robust 0–1 knapsack problem to systematically analyze fixed-parameter tractability under several natural parameters. The main contributions include showing that the problem is XP but W[1]-hard with respect to the uncertainty budget Γ; it remains NP-hard even with two distinct due dates, yet becomes pseudopolynomially solvable when the number of distinct due dates is fixed; it is polynomially solvable under a common due date; and the number of jobs with non-zero deviation is proven to be a fixed-parameter tractable (FPT) parameter.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this study, we investigate a robust single-machine scheduling problem under processing time uncertainty. The uncertainty is modeled using the budgeted approach, where each job has a nominal and deviation processing time, and the number of deviations is bounded by Gamma. The objective is to minimize the maximum number of tardy jobs over all possible scenarios. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that some natural parameter of the problem is bounded by a constant. We consider three parameters: the robustness parameter Gamma, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parametrized-complexity theory, we prove that the problem is W[1]-hard with respect to Gamma, but can be solved in XP time with respect to the same parameter. With respect to the number of different due dates, we establish a stronger hardness result by showing that the problem remains NP-hard even when there are only two different due dates and is solvable in pseudo-polynomial time when the number of due dates is upper bounded by a constant. To complement these results, we show that the case of a common (single) due date, reduces to a robust binary knapsack problem with equal item profits, which we prove to be solvable in polynomial time. Finally, we prove that the problem is solvable in FPT time with respect to the number of nonzero deviations.

Problem

• robust scheduling
• budgeted uncertainty
• single-machine scheduling
• tardy jobs
• parameterized complexity

Innovation

• parameterized complexity
• budgeted uncertainty
• robust scheduling
• fixed-parameter tractability
• tardy jobs