The streaming model was introduced to parameterized complexity independently by Fafianie and Kratsch [MFCS14] and by Chitnis, Cormode, Hajiaghayi and Monemizadeh [SODA15]. Subsequently, it was broadened by Chitnis, Cormode, Esfandiari, Hajiaghayi and Monemizadeh [SPAA15] and by Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh and Vorotnikova [SODA16]. Despite its strong motivation, the applicability of the streaming model to central problems in parameterized complexity has remained, for almost a decade, quite limited. Indeed, due to simple $\Omega(n)$-space lower bounds for many of these problems, the $k^{O(1)}\cdot {\rm polylog}(n)$-space requirement in the model is too strict. Thus, we explore {\em semi-streaming} algorithms for parameterized graph problems, and present the first systematic study of this topic. Crucially, we aim to construct succinct representations of the input on which optimal post-processing time complexity can be achieved. - We devise meta-theorems specifically designed for parameterized streaming and demonstrate their applicability by obtaining the first $k^{O(1)}\cdot n\cdot {\rm polylog}(n)$-space streaming algorithms for well-studied problems such as Feedback Vertex Set on Tournaments, Cluster Vertex Deletion, Proper Interval Vertex Deletion and Block Vertex Deletion. In the process, we demonstrate a fundamental connection between semi-streaming algorithms for recognizing graphs in a graph class H and semi-streaming algorithms for the problem of vertex deletion into H. - We present an algorithmic machinery for obtaining streaming algorithms for cut problems and exemplify this by giving the first $k^{O(1)}\cdot n\cdot {\rm polylog}(n)$-space streaming algorithms for Graph Bipartitization, Multiway Cut and Subset Feedback Vertex Set.

翻译：流模型由Fafianie和Kratsch [MFCS14] 以及Chitnis、Cormode、Hajiaghayi和Monemizadeh [SODA15] 独立引入参数化复杂度领域。随后，Chitnis、Cormode、Esfandiari、Hajiaghayi和Monemizadeh [SPAA15] 以及Chitnis、Cormode、Esfandiari、Hajiaghayi、McGregor、Monemizadeh和Vorotnikova [SODA16] 对其进行了拓展。尽管该模型具有强大的动机，但在近十年间，流模型在参数化复杂度核心问题上的可应用性仍然相当有限。事实上，由于许多此类问题存在简单的$\Omega(n)$空间下界，模型中对$k^{O(1)}\cdot {\rm polylog}(n)$空间的需求过于严格。因此，我们探究参数化图问题的半流算法，并首次对该主题进行系统性研究。关键目标是构建输入的精简表示，以实现最优后处理时间复杂度。具体而言： - 我们设计了专用于参数化流算法的元定理，并通过为广受研究的问题（如锦标赛反馈顶点集、团簇顶点删除、真区间顶点删除和块顶点删除）构建首个$k^{O(1)}\cdot n\cdot {\rm polylog}(n)$空间流算法，证明了这些元定理的可应用性。在此过程中，我们揭示了识别图类H中图的半流算法与顶点删除到H问题的半流算法之间的基本联系。 - 我们提出了一套用于获取割问题流算法的算法机制，并通过为图二分性、多路割和子集反馈顶点集问题构建首个$k^{O(1)}\cdot n\cdot {\rm polylog}(n)$空间流算法进行了例证。