On the Recognition of Strong-Robinsonian Incomplete Matrices

Abstract

A matrix is incomplete when some of its entries are missing. A Robinson incomplete symmetric matrix is an incomplete symmetric matrix whose non-missing entries do not decrease along rows and columns when moving toward the diagonal. A Strong-Robinson incomplete symmetric matrix is an incomplete symmetric matrix A such that a k,l ≥ a i,j if a i,j and a k,l are two non-missing entries of A and i ≤ k ≤ l ≤ j. On the other hand, an incomplete symmetric matrix is Strong-Robinsonian if there is a simultaneous reordering of its rows and columns that produces a Strong-Robinson matrix. In this document, we first show that there is an incomplete Robinson matrix which is not Strong-Robinsonian. Therefore, these two definitions are not equivalent. Secondly, we study the recognition problem for Strong-Robinsonian incomplete matrices. It is known that recognition of incomplete Robinsonian matrices is NP-Complete. We show that the recognition of incomplete Strong-Robinsonian matrices is also NP-Complete. However, we show that recognition of Strong-Robinsonian matrices can be parametrized with respect to the number of missing entries. Indeed, we present an O(|w| b n 2 ) recognition algorithm for Strong-Robinsonian matrices, where b is the number of missing entries, n is the size of the matrix, and |w| is the number of different values in the matrix.

Modules in Robinson Spaces

“…We assume that any compatible order < i of a separable copoint C i defines an admissible bipartition {C ′ i , C ′′ i } of C i as defined in Subsection 4.2. The total order < is defined as follows: for two points x, y of X we set x < y if and only if (1) x ∈ α, y ∈ β for two different points α, β ∈ C * p and α < * β or (2)…”

“…Laurent, Seminaroti and Tanigawa [27] presented a characterization of Robinson matrices in terms of forbidden substructures, extending the notion of asteroidal triples in graphs to weighted graphs. More recently, Aracena and Thraves Caro [1] presented a parametrized algorithm for the NP-complete problem of recognition of Robinson incomplete matrices. Armstrong et al [2] presented an optimal O(n 2 ) time algorithm for the recognition of strict circular Robinson dissimilarities (Hubert et al [21] defined circular seriation first and it was studied also in the papers [31] and [23]).…”