Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

TL;DR

This paper introduces novel fixed-parameter algorithms for Euclidean Distance Matrix Completion, leveraging structural patterns and proximity to triviality to achieve tractability in cases previously considered hard.

Contribution

It presents the first fixed-parameter algorithms for d-EDMC based on structural parameters and a polynomial-time solution for fixed d and fill-in, advancing the understanding of EDM completion complexity.

Findings

First FPT algorithm parameterized by d and unspecified entries per row/column.

First FPT algorithm parameterized by d and covering principal submatrices.

Polynomial-time algorithm for fixed d and fill-in, combining geometry and algebraic tools.

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.