组合可离散距离几何问题的秩计数理论
A Rank-Count Theory for the Combinatorial Discretizable Distance Geometry Problem
AI总结：
针对组合可离散距离几何问题，研究人员开发代数秩计数理论，证明在镜像分离参数且存在可行参考解时，其可行二进制分支码构成\n\n上的仿射空间，解决了额外约束下的拓扑解计数难题。
AI中文摘要：
距离几何问题（DGP）要求在\mathbb{R}^K\n中对具有\n\n个顶点的加权图进行几何实现，使得顶点间的欧氏距离匹配给定的边权重。当输入包含这样的顶点顺序：每个非种子顶点都有\n\n个构成团的前驱顶点时，可通过\n\n次 lateration（K- lateration）将搜索空间离散化，分支为二叉树。这个被称为组合可离散距离几何问题（Combinatorial DDGP）的子类，目标是确定满足所有距离约束的实现数量。在无额外距离约束时，实现数量几乎总是\n\n，而在存在额外约束时的拓扑解计数一直难以捉摸。我们针对可行的二进制分支码开发了代数秩计数理论，证明当存在可行参考解且参数为镜像分离时，这些二进制分支码构成\n\n上的仿射空间。
英文摘要：
The Distance Geometry Problem (DGP) asks for a geometric realization of a weighted graph with \(n\) vertices in \(\mathbb{R}^K\) such that Euclidean distances between vertices match the given edge weights. When a vertex order where every non-seed vertex has \(K\) predecessors inducing a clique is part of the input, the search space can be discretized via \(K\)-lateration, branching into a binary tree. In this subclass, known as the Combinatorial Discretizable Distance Geometry Problem (Combinatorial DDGP), the goal is to determine the number of realizations satisfying all distance constraints. While the number of realizations is almost always \(2^{n-K}\) when no additional distance constraints are present, a topological solution count in the presence of additional constraints has remained elusive. We develop an algebraic rank-count theory for the feasible binary branch codes, proving that under mirror-separated parameters they form an affine space over \(\mathbb{F}_2\) whenever a viable reference solution exists.
