Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful approach for this is the construction of an extended formulation, which is a linear program in a higher dimensional space whose projection yields the original linear program. For many problems it is known that a small extended formulation cannot exist. However, most of these problems are either $\mathsf{NP}$-hard (like TSP), or only quite complicated polynomial time algorithms are known for them (like for the matching problem). In this work we study the minimum makespan problem on identical machines in which we want to assign a set of $n$ given jobs to $m$ machines in order to minimize the maximum load over the machines. We prove that the canonical formulation for this problem has extension complexity $2^{Ω(n/\log n)}$, even if each job has size 1 or 2 and the optimal makespan is 2. This is a case that a trivial greedy algorithm can solve optimally! For the more powerful configuration integer program, we even prove a lower bound of $2^{Ω(n)}$. On the other hand, we show that there is an abstraction of the configuration integer program admitting an extended formulation of size $f(\text{opt})\cdot \text{poly}(n,m)$. In addition, we give an $O(\log n)$-approximate integral formulation of polynomial size, even for arbitrary processing times and for the far more general setting of unrelated machines.
A Note on Hardness of Multiprocessor Scheduling with Scheduling Solution Space Tree
研究了在一组相同的并行处理机上以最小化完工时间为目标的独立工件列表的非抢占式调度问题的计算复杂性。通过定义\textit{Scheduling Solution Space Tree (SSST)}的数据结构，我们首次尝试探索显示问题的穷举解空间的组合结构。的\textit{SSST}的属性被正式定义，并通过我们的分析结果的特点。我们开发了一种独特的技术来显示使用SSST和\textit{Weighted Scheduling Solution Space Tree (WSSST)}数据结构的问题$\mathcal{NP}$。我们设计了第一个非确定性多项式时间算法\textit{Magic Scheduling (MS)}的基础上减少框架的问题。我们还定义了一个新的变种，多处理器调度包括用户作为一个额外的输入参数。我们正式建立的复杂性类的变体的归约原则。最后，我们通过探讨未来研究调查的几个有趣的开放性问题来结束本文。
A Note on NP-Hardness of Preemptive Mean Flow-Time Scheduling for Parallel Machines
Baptiste，Brucker，Chrobak，Dürr，Kravchenko和Sourd在论文“The complex of mean flow time scheduling problems with release times”中，证明了调度问题$P|pmtn,r_j|\sum C_j$的强NP-困难性，即以最小化平均流时间为目标的多处理器抢占式调度。指出了他们证明中的一个严重错误，并给出了该问题强NP-困难性的一个新证明。
Parameterized Complexity of Scheduling Problems in Robotic Process Automation
Czech Technical University in Prague（布拉格捷克技术大学）；Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague（捷克技术大学捷克斯洛伐克信息学、机器人学与自动化研究所）；Sorbonne University（索邦大学）；CNRS（法国国家科学研究中心）
