Single-machine scheduling with fixed energy recharging times to minimize the number of late jobs and the number of just-in-time jobs: A parameterized complexity analysis

Abstract

We study single-machine scheduling problems where processing each job requires both processing time and rechargeable energy. Subject to a predefined energy capacity, energy can be recharged after each job during a fixed recharging period. Our focus is on two due date-related scheduling criteria: minimizing the number of late jobs and maximizing the weighted number of jobs completed exactly at their due dates. This study aims to analyze the parameterized tractability of the two problems and develop fixed-parameter algorithms with respect to three natural parameters: the number of different due dates $v_d$, the number of different processing times $v_p$, and the number of different energy consumptions $v_e$. Following the proofs of $\mathrm{NP}$-hardness across several contexts, we demonstrate that both problems remain intractable when parameterized by $v_d$ and $v_p$. To complement our results, we show that both problems become fixed-parameter tractable (FPT) when parameterized by $v_e$ and $v_d$, and are solvable in polynomial time when both $v_e$ and $v_p$ are constant.