AI 中文总结
该研究证明单位长度唯一机器优先调度（UMPS）无法在常数因子及多项式对数因子内近似，其难度下界可传递至相关带通信延迟的调度模型。
AI 中文摘要
由[DKRSTZ22]提出的唯一机器优先调度（Unique-Machine Precedence Scheduling, UMPS）问题，旨在为每个作业仅有一台合格机器的带优先约束作业，寻找使总完成时间（makespan）最小的调度方案。一方面，UMPS通过允许优先图为任意有向无环图（DAG）而非不相交链的并集，推广了作业车间调度问题；另一方面，UMPS可将保持近似性的归约应用于带通信延迟的调度问题，包括作业-作业延迟模型[DKRSTZ22]和作业-机器延迟模型[RSY23]。尽管UMPS具有核心作用，但其近似性仍未被充分理解：即使对于单位长度作业，已知的调度技术似乎也无法产生非平凡的近似比，且[DKRSTZ22]提出的是否存在多项式对数近似算法的问题仍未解决。在难度方面，单位长度作业的此前最佳下界仅为从作业车间调度继承而来的5/4[WHHHLSS97]。我们证明，单位长度UMPS是NP难的，无法在任何常数因子内近似。我们进一步证明，假设NP不在准多项式时间内，单位长度UMPS不存在多项式时间的(log n)^γ近似算法，其中γ>0为某常数。通过UMPS的已知归约，这些下界也可传递到对应的单位长度通信延迟调度模型。我们的证明通过从超图着色承诺问题进行归约得到：在“是”实例中，输入超图存在平衡着色；在“否”实例中，超图无大独立集。利用[GL18]的超图着色难度实例化该归约，可得到任意常数因子的不可近似性；结合[GHHSV17]的4-均匀超图着色（4-可着色）难度与超图的某种复合操作，可得到多项式对数因子的不可近似性。
英文摘要
The Unique-Machine Precedence Scheduling (UMPS) problem, introduced by [DKRSTZ22], seeks a makespan-minimizing schedule of precedence-constrained jobs when each job has a unique eligible machine. On the one hand, UMPS generalizes job shop scheduling by allowing the precedence graph to be an arbitrary DAG rather than a disjoint union of chains. On the other hand, UMPS admits approximation-preserving reductions to scheduling problems with communication delays, including the job-job delay model [DKRSTZ22] and the job-machine delay model [RSY23]. Despite its central role, the approximability of UMPS has remained poorly understood: even for unit-length jobs, known scheduling techniques do not seem to yield a non-trivial approximation, and the existence of a polylogarithmic approximation was left open by [DKRSTZ22]. On the hardness side, the previous best lower bound for unit-length jobs was only the 5/4 inherited from job shop scheduling [WHHHLSS97]. We prove that unit-length UMPS is NP-hard to approximate within any constant factor. We further show that, assuming NP is not in quasi-polynomial time, unit-length UMPS admits no polynomial-time $(\log n)^γ$-approximation for some constant $γ>0$. Via the known reductions from UMPS, these lower bounds also transfer to the corresponding unit-length communication-delay scheduling models. Our proof proceeds via a reduction from a hypergraph coloring promise problem. In the yes case, the input hypergraph admits a balanced coloring, while in the no case, the hypergraph has no large independent set. Instantiating this reduction with the hardness of [GL18] gives arbitrary constant-factor inapproximability, while combining the $4$-colorable $4$-uniform hypergraph coloring hardness of [GHHSV17] with a certain composition operation for hypergraphs yields the polylogarithmic factor inapproximability.
