TCS Journal 2026 Journal Article
Parameterized algorithms and complexity for scheduling with precedence constraints and time windows
- Feng Shi
- Na Feng
- Yicong Zhu
- Jingyi Liu
- Jianxin Wang
- Guangwei Wu
In the paper, we study several variants of the Scheduling problem with precedence constraints and time windows defined on a task set S, denoted by P∣prec, ri, di ∣γ, and present improved fixed-parameter algorithms parameterized by the maximum processing time p max and the maximum number μ of overlapping time windows, defined as μ = max t ∈ N | { i ∈ S ∣ r i ≤ t < d i } |. Firstly, we propose an algorithm for P∣prec, ri, di ∣C max with time complexity O ( ( p max + 2 ) μ p max n 3 ), where the aim is to find a feasible schedule minimizing the makespan C max, and n is the number of tasks. This significantly improves the previously best-known algorithm for P∣prec, ri, di ∣⋆ with time complexity O ( p max 2 μ · 16 μ μ · n 3 ), where ⋆ indicates that the variant has no objective function and just decides the existence of a feasible schedule. Next, for the unit processing time case P ∣ p r e c, p i = 1, r i, d i ∣ C max, we develop an algorithm with time complexity O(2 μ μmn 3), where m is the number of machines, improving the previously best-known algorithm for P ∣ p r e c, p i = 1, r i, d i ∣ ★ with time complexity O(16 μ n 4). Then, we extend the two algorithms to the typed machine setting, i. e. , P ∣ M j ( t y p e ), p r e c, r i, d i ∣ C max and P ∣ M j ( t y p e ), p r e c, p i = 1, r i, d i ∣ C max, as well as to variants with other objective functions such as the maximum flow time F max and mean weighted flow time F mean. Finally, we show that the problem P ∣ o u t t r e e, p i = 1, d i ∣ ★, where the precedence constraints are restricted to an outtree, is para-NP-hard when parameterized by the maximum vertex out-degree Δ of the outtree.