Uniform Embeddings for Robinson Similarity Matrices

TL;DR

This paper investigates the problem of uniformly embedding Robinson similarity matrices with respect to multiple thresholds, providing a necessary and sufficient condition and an efficient algorithm for the case of two thresholds.

Contribution

It introduces the concept of uniform embedding for Robinson matrices with multiple thresholds and offers a characterization and an algorithm for the case of two thresholds.

Findings

Characterizes when a Robinson matrix admits a uniform embedding.

Provides a combinatorial algorithm for the case of two thresholds.

Establishes a necessary and sufficient condition based on associated graph paths.

Abstract

A Robinson similarity matrix is a symmetric matrix where the entry values on all rows and columns increase toward the diagonal. Decompose the Robinson matrix into the sum of k {0, 1}-matrices, then these k {0, 1}-matrices are the adjacency matrices of a set of nested unit interval graphs. Previous studies show that unit interval graphs coincide with indifference graphs. An indifference graph has an embedding that maps each vertex to a real number, where two vertices are adjacent if their embedding is within a fixed threshold distance. In this thesis, consider k different threshold distances, we study the problem of finding an embedding that, simultaneously and with respect to each threshold distance, embeds the k indifference graphs corresponding to the k adjacency matrices. This is called a uniform embedding of a Robinson matrix with respect to the k threshold distances. We give a sufficient and necessary condition on Robinson matrices that have a uniform embedding, which is derived from paths in an associated graph. We also give an efficient combinatorial algorithm to find a uniform embedding or give proof that it does not exist, for the case where k = 2.