A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty
Noam Goldberg, Dvir Shabtay

TL;DR
This paper analyzes the computational complexity of a robust scheduling problem under processing time uncertainty using parametrized complexity theory, identifying tractability and hardness results for various parameters.
Contribution
It provides a detailed complexity analysis of the problem, showing W[1]-hardness for certain parameters and polynomial or FPT algorithms for others.
Findings
Problem is W[1]-hard with respect to the robustness parameter Gamma.
Problem is solvable in XP time with respect to Gamma.
Special case with a common due date reduces to a polynomial-time solvable robust binary knapsack.
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…
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Taxonomy
TopicsScheduling and Optimization Algorithms · Risk and Portfolio Optimization · Optimization and Search Problems
