组合数据上的组合算法：判定预层上的层
Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves
AI总结：
本研究采用范畴论视角，将图上动态规划直觉推广到广义数学结构，证明结构化分解定义Grothendieck拓扑，层表示的问题在有界分解输入上可线性时间判定，并在AlgebraicJulia中实现。
摘要：
算法研究者都清楚，在组合（甚至递归）图上进行计算时，快速动态规划算法往往是最优选择。本文首次研究如何将这一民间直觉推广到更广泛的数学结构中。我们通过采用范畴论视角实现这种横向通用性，并证明两点：（1）结构化分解（近期对多种图分解的抽象推广）在数据范畴（粘合范畴）上定义了Grothendieck拓扑；（2）任何能表示为相对于这些拓扑的层的计算问题，在允许有界宽度分解、且分解形状的反馈顶点数有界的输入类别上，都可以在线性时间内判定。这一结论可直接推导出任意C-集范畴对象上的算法；这类结构包括但不限于：对称图、有向图、有向多重图、超图、有向超图、数据库、单纯复形、圆形端口图和半边图。由此我们搭建起层论、结构图论与参数化复杂性理论工具之间的桥梁；我们认为这对构建通用的动态规划算法代数理论是一种极具成效的方法。最后，我们在AlgebraicJulia生态系统中实现了主要算法贡献的具体实现，以支撑理论结果。
英文摘要：
Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to mathematical structures writ large. We achieve this horizontal generality by adopting a categorial perspective which allows us to show that: (1) structured decompositions (a recent, abstract generalization of many graph decompositions) define Grothendieck topologies on categories of data (adhesive categories) and that (2) any computational problem which can be represented as a sheaf with respect to these topologies can be decided in linear time on classes of inputs which admit decompositions of bounded width and whose decomposition shapes have bounded feedback vertex number. This immediately leads to algorithms on objects of any C-set category; these include -- to name but a few examples -- structures such as: symmetric graphs, directed graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases, simplicial complexes, circular port graphs and half-edge graphs. Thus we initiate the bridging of tools from sheaf theory, structural graph theory and parameterized complexity theory; we believe this to be a very fruitful approach for a general, algebraic theory of dynamic programming algorithms. Finally we pair our theoretical results with concrete implementations of our main algorithmic contribution in the AlgebraicJulia ecosystem.
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