On the Recognition of Strong-Robinsonian Incomplete Matrices
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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|^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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