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arXiv · 2101.03033

On the Recognition of Strong-Robinsonian Incomplete Matrices

Julio Aracena·Christopher Thraves Caro
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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 $a_{k,l} \geq a_{i,j}$ if $a_{i,j}$ and $a_{k,l}$ are two non-missing entries of $A$ and $i\leq k \leq l \leq j$. 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 $O(|w|^bn^2)$ 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.

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Julio Aracena, Christopher Thraves Caro. 2021-01-08. On the Recognition of Strong-Robinsonian Incomplete Matrices. https://arxiv.org/abs/2101.03033

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