Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

2026-03-19

AI Summary

This study addresses the d-dimensional Euclidean Distance Matrix Completion (d-EDMC) problem: determining whether a partially specified Euclidean distance matrix admits an exact completion in ℝᵈ. The authors propose a novel parameterized framework based on the notion of “distance from triviality” and present the first exact—rather than approximate—algorithm for this problem. Their main contributions include two fixed-parameter tractable (FPT) algorithms, parameterized respectively by the maximum number of missing entries per row/column and by the minimum number of fully specified principal submatrices needed to cover all specified entries. Additionally, they establish a polynomial-time algorithm when both the dimension d and the graph’s fill-in number are fixed. The approach integrates techniques from distance geometry, real algebraic geometry, chordal graph theory, and a new compression method to reduce instances to constant size, while also revealing new connections between d-EDMC and graph-theoretic problems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Euclidean Distance Matrix Completion
• d-EDMC
• matrix completion
• computational geometry
• distance geometry

Innovation

• Euclidean Distance Matrix Completion
• fixed-parameter tractability
• distance-from-triviality
• matrix compression
• chordal graph