In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: (1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on O(1) rows (for fixed values of the parameters). (2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. (3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.

翻译：在d维欧几里得距离矩阵补全（d-EDMC）问题中，目标是判断给定的部分成对距离矩阵能否扩展为d维空间中的完整欧几里得距离矩阵。该问题是计算几何领域的基石，具有广泛的应用价值。传统研究多利用与半定规划的联系，通常得到近似算法；而本文聚焦于精确算法，提出新颖的“距平凡性距离”参数化框架，为d-EDMC问题建立可解性结果。我们识别出输入数据中体现条目密度的关键结构模式，包括弦子结构及由完全指定的主子矩阵覆盖指定条目。主要成果包括：（1）首个以d和每行/列未指定条目最大数为参数的d-EDMC固定参数算法（FPT算法），该算法通过新颖的压缩技术将实例简化为O(1)行子矩阵（参数固定时）；（2）首个以d和能覆盖所有指定条目的完全指定主子矩阵最小数量为参数的d-EDMC固定参数算法，同样通过压缩算法实现；（3）当d和表示指定条目的自然图的最小填充数均为固定常数时，给出多项式时间算法，该成果结合了距离几何工具与实代数几何算法。本文揭示了EDM补全与图问题间的有趣关联，所提算法融合了两领域的技术方法。