We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximise the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalises to multiple in/output vertices, as well as regular constraints.

翻译：我们提出了一种新的代数技术，可在多项式空间内解决序列流问题。该任务旨在最大化通过图的流量，其中边容量可通过从给定有限集合中选择容量标记序列随时间改变。我们的方法基于有限半群的一个新颖分解定理，该定理应用于合适的流半群，能够推导出小型见证。该方法可推广至多输入/输出顶点以及正则约束情形。