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On $\ell_1$ embeddings of finite metric spaces, and sphere-of-influence graphs

Stanislav Jabuka, Ehsan Mirbagheri

TL;DR

The paper introduces the pair-cut cone $PCUT_n$, a focused subcone of the cut cone generated by pair-cut metrics, and provides a complete, direct inequality-based characterization for membership in $PCUT_n$ via star-traces and the overall trace of the metric. It then analyzes the square cut-matrix $S_{sq}$ to give an explicit solution formula for the pair-cut coefficients and derives a concrete, testable criterion: a metric lies in $PCUT_n$ iff the inequalities $s_i+s_j \ge (n-4)d(i,j) + \frac{2}{n-2}\mathrm{Tr}(d)$ hold for all pairs $\{i,j\}$. The authors connect these results to the full cut cone $CUT_n$, providing a separate sufficient condition for $CUT_n$ membership and a framework based on the full-cut matrix $S$ and its right inverse. They apply these tools to the study of sphere-of-influence graphs (SIGs), proving that there exist graphs (notably stars) with no SIG-metric in $PCUT_n$, thereby highlighting limitations of $PCUT_n$ for SIG-embeddability; they also explore multiple SIG-metrics on graphs, showing that membership can vary between metrics. Finally, the paper develops a constructive analysis of the full cut matrix, including a right inverse and an explicit kernel basis, outlining a practical, albeit computationally challenging, approach to determining CUT_n membership via kernel adjustments.

Abstract

We introduce the {\em pair-cut cone $PCUT_n$} of metrics on sets with $n\ge 3$ elements, that correspond to linear combinations with non-negative coefficients of the cut-metrics resulting from cuts that are pairs. Given a metric, we fully characterize membership in the pair-cut cone in terms of quantities computed from the metric directly. We also prove a new result by which a metric $d$ that satisfies a system of inequalities, lies in the (full) cut cone of metrics, making it $\ell_1$-embeddable into Euclidean space. We give applications of our results to the $\ell_1$-embeddability of simple graphs into Euclidean space as {\em sphere-of-influence graphs}. We exhibit an example of a simple graph that admits no such $\ell_1$-metric in the pair-cut cone.

On $\ell_1$ embeddings of finite metric spaces, and sphere-of-influence graphs

TL;DR

The paper introduces the pair-cut cone , a focused subcone of the cut cone generated by pair-cut metrics, and provides a complete, direct inequality-based characterization for membership in via star-traces and the overall trace of the metric. It then analyzes the square cut-matrix to give an explicit solution formula for the pair-cut coefficients and derives a concrete, testable criterion: a metric lies in iff the inequalities hold for all pairs . The authors connect these results to the full cut cone , providing a separate sufficient condition for membership and a framework based on the full-cut matrix and its right inverse. They apply these tools to the study of sphere-of-influence graphs (SIGs), proving that there exist graphs (notably stars) with no SIG-metric in , thereby highlighting limitations of for SIG-embeddability; they also explore multiple SIG-metrics on graphs, showing that membership can vary between metrics. Finally, the paper develops a constructive analysis of the full cut matrix, including a right inverse and an explicit kernel basis, outlining a practical, albeit computationally challenging, approach to determining CUT_n membership via kernel adjustments.

Abstract

We introduce the {\em pair-cut cone } of metrics on sets with elements, that correspond to linear combinations with non-negative coefficients of the cut-metrics resulting from cuts that are pairs. Given a metric, we fully characterize membership in the pair-cut cone in terms of quantities computed from the metric directly. We also prove a new result by which a metric that satisfies a system of inequalities, lies in the (full) cut cone of metrics, making it -embeddable into Euclidean space. We give applications of our results to the -embeddability of simple graphs into Euclidean space as {\em sphere-of-influence graphs}. We exhibit an example of a simple graph that admits no such -metric in the pair-cut cone.

Paper Structure

This paper contains 24 sections, 22 theorems, 133 equations, 5 figures.

Table of Contents

  1. Introduction and Main Results
  2. $\ell_1$-embeddability and the pair-cut cone
  3. $\ell_1$-embeddability and the cut cone
  4. Sphere-of-Influence Graphs
  5. Organization
  6. Background
  7. Finite Metric spaces
  8. The Cut Cone $CUT_n$
  9. Sphere-of-Influence Graphs
  10. The Square Cut-Matrix and the Pair-Cut Cone $PCUT_n$
  11. The pair-cut cone
  12. The square cut-matrix
  13. The spectral decomposition of $S_{sq}$
  14. The spectral decomposition of $S_{sq}$ and $S_{sq}^{-1}$
  15. Solving the system $S_{sq}w = d$
  16. ...and 9 more sections

Key Result

Theorem 1.1

For a metric $d$ on $V_n$ with $n\ge 5$, define the trace Tr$(d)$ of the metric $d$, and the star-trace $s_i$ of a vertex $i\in V_n$ as: Then $d$ lies in the pair-cut cone $PCUT_n$ if and only if

Figures (5)

  • Figure 1: The star graph $S(n)$ shown here with $n=16$. The central vertex is labeled $0$ while the peripheral vertices are enumerated by the elements of $V_n=\{1,\dots, n\}$.
  • Figure 2: A visualization of $CUT_3^\square$ and $PCUT^\square_3$. The former is the tetrahedron spanned by $\delta_{\{\}}$, $\delta_{\{1,2\}}$, $\delta_{\{1,3\}}$ and $\delta_{\{2,3\}}$, while the latter is the triangle spanned by $\delta_{\{1,2\}}$, $\delta_{\{1,3\}}$ and $\delta_{\{2,3\}}$.
  • Figure 3: The star graph $S(3)$.
  • Figure 4: A graph with 7 vertices.
  • Figure 5: The star graph $S(n)$ shown here with $n=16$. The central vertex is labeled $0$ while the peripheral vertices are enumerated by the elements of $V_{16}=\{1,\dots, 16\}$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Example 1.4
  • Theorem 1.5
  • Example 2.1
  • Theorem 2.2: Carathéodory Caratheodory
  • Theorem 2.3
  • Theorem 2.4: Avis-Deza AvisDeza
  • Theorem 2.5
  • ...and 37 more