When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations
Bentert, Matthias, Fomin, Fedor V., Golovach, Petr A., Ramanujan, M. S., Saurabh, Saket · arXiv (Cornell University) · 2025
Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in ℝ^d (d ≥ 1), or equivalently, distance spaces that can be isometrically embedded in ℝ^d. In this work, we investigate whether a distance space can be isometrically embedded in ℝ^d after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing, asks whether an input distance space on n points can be transformed, using at most k modifications, into a space that is isometrically embeddable in ℝ^d. We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves Euclidean Embedding Editing in time (dk)^𝒪(d+k) + n^𝒪(1). The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of Euclidean Embedding Editing with 𝒪((dk)²) points. For the special but important case of Euclidean Embedding Editing where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of min{(d+3)^k, 2^{d+k}} ⋅ n^𝒪(1). Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most 2 ⋅ Opt outliers in time 2^d ⋅ n^{𝒪(1)}. This 2-approximation algorithm improves upon the previous (3+ε)-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations.