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Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, Saket Saurabh

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.

Algorithms for Euclidean Distance Matrix Completion: Exploiting Proximity to Triviality

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.
Paper Structure (8 sections, 15 theorems, 2 equations, 3 figures)

This paper contains 8 sections, 15 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our results and methods
  3. Distance from Triviality I
  4. Distance from Triviality II
  5. Distance from Triviality III
  6. Related Work
  7. Preliminaries
  8. Compressing $G$-partial matrices when $G$ is dense (distance from triviality I and II)

Key Result

Theorem 1

$\forall\varepsilon\in(0,1)$, $d$-EDMC remains strongly $\mathop{\mathrm{NP}}\nolimits$-hard even for instances $(M,d)$ with $d=2$ in which the $n\times n$ matrix $M$ contains at most $\varepsilon n$ unspecified entries.

Figures (3)

  • Figure 1: Top: A $9 \times 9$ partial matrix $M$ that excludes $4$-block pattern. It contains a $3$-block pattern formed by rows $\{3, 8, 9\}$ and columns $\{1, 5, 7\}$. Bottom Left: the graph $G$ underlying the partial EDM matrix $M$. The adjacencies of $G$ are formed by the unspecified entries in the matrix. Bottom Right: the complement graph $\overline{G}$ induced by the unspecified entries of $M$. The red edges of $\overline{G}$ form $K_{3,3}$, however $\overline{G}$ does not contain $K_{4,4}$.
  • Figure 2: A partial EDM matrix whose underlying graph is chordal.
  • Figure 3: Graph $G$, $G$-partial matrix $A$, and EDM completion $D$ of $A$. The distance space ${{\cal X}}$ defined by $D$ is embeddable into $\mathbb{R}^2$. For example, a realization $\varphi$ of ${{\cal X}}$ would map the elements of ${{\cal X}}$ to points $p_1= (0,0)$, $p_2= (1,0)$, $p_3= (0.5,1)$, and $p_4= (0.5,2)$.

Theorems & Definitions (19)

  • Theorem 1
  • Definition 2: $t$-block pattern
  • Example 3
  • Theorem 4: Compression for $K_{t,t}$-free complements
  • Theorem 5
  • Corollary 5
  • Remark 6: Generality of \ref{['thm:compression']} and \ref{['cor:fptKtt']}
  • Theorem 7
  • Corollary 7
  • Remark 8
  • ...and 9 more