Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

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.

💡 Research Summary

The paper investigates the Euclidean Distance Matrix Completion problem (d‑EDMC), where one is given a symmetric n × n matrix M with some entries missing and a target dimension d, and must decide whether the missing entries can be filled so that the resulting matrix is a Euclidean distance matrix realizable in ℝ^d. While earlier work largely focused on semidefinite programming relaxations and approximation, this work adopts a parameterized‑complexity viewpoint, introducing the notion of “distance‑from‑triviality”: the distance of an instance from the trivial case where the matrix is completely specified (which can be checked in polynomial time).

Three families of structural parameters are studied, each capturing a different way of being “close” to the trivial case.

1. Maximum number of unspecified entries per row/column (t).
   The authors show that if the complement graph of the specified entries does not contain a K_{t,t} (equivalently, the matrix excludes a t‑block of all‑unspecified entries), then the instance can be compressed in polynomial time to a principal submatrix of size (d + 1)·O(t²). This compression (Theorem 4) either solves the problem directly or yields an equivalent smaller instance. By invoking a real‑algebraic‑geometry decision procedure (Theorem 5) on the compressed matrix, they obtain an FPT algorithm running in time 2^{O((d+1)·t²)}·poly(n) (Corollary 6). Thus (d, t) is a fixed‑parameter tractable parameterization.

2. Maximum number of unspecified entries per row (Δ).
   When each row contains at most Δ missing entries, the underlying graph has maximum degree Δ. The authors refine the compression bound to (d + 1)(Δ + 1)² (Theorem 8) and, combined with the same algebraic decision routine, achieve a running time 2^{O(d²·Δ⁴·log(dΔ))}·poly(n) (Corollary 9). This gives another FPT result parameterized by (d, Δ).

3. Coverability by a small number of fully specified principal submatrices (k).
   If all specified entries lie inside k fully specified principal submatrices, the underlying graph admits an edge‑clique cover of size k. Theorem 11 shows that the instance can be reduced to a principal submatrix of size (d + 1)·k². After applying the algebraic decision algorithm, the total running time becomes 2^{O(d²·k⁴·log(dk))} + 2^{O(k)}·poly(n) (Corollary 12). This parameterization is independent of any forbidden bipartite subgraph condition.

In addition to these three main results, the paper presents a polynomial‑time algorithm for the case where the complement graph has a fixed minimum fill‑in (i.e., the graph becomes chordal after adding a constant number of edges). By exploiting chordality and the fact that EDM realizability can be checked locally on each maximal clique, the authors obtain a deterministic polynomial‑time procedure for this class.

The technical core of the compression results is the identification of an “irrelevant element”: a row/column that can be safely removed without changing the answer. The existence of such an element is guaranteed by structural properties (K_{t,t}‑freeness, bounded degree, or the presence of a large clique in an edge‑clique cover). Repeated removal yields a kernel whose size depends only on the chosen parameters, although the kernel is not a classic parameterized kernel because the numeric values of the remaining entries may still depend on the original instance.

The paper also establishes strong NP‑hardness even when the matrix is dense: for any ε ∈ (0,1) the problem remains NP‑hard when at most εn entries are missing, showing that mere sparsity does not help unless additional structure is present.

Overall, the work bridges distance geometry and graph theory, demonstrating that by measuring how far a partial EDM is from the fully specified case, one can obtain fixed‑parameter tractable and even polynomial‑time algorithms for several natural structural parameters. This opens a new avenue for exact algorithms in distance‑based reconstruction problems and suggests that similar “distance‑from‑triviality” parameterizations could be fruitful for other geometric completion tasks.