Complexity of Combinatorial Matrix Completion With Diameter Constraints
Abstract
We thoroughly study a novel and still basic combinatorial matrix completion problem: Given a binary incomplete matrix, fill in the missing entries so that the resulting matrix has a specified maximum diameter (that is, upperbounding the maximum Hamming distance between any two rows of the completed matrix) as well as a specified minimum Hamming distance between any two of the matrix rows. This scenario is closely related to consensus string problems as well as to recently studied clustering problems on incomplete data. We obtain an almost complete complexity dichotomy between polynomialtime solvable and NPhard cases in terms of the minimum distance lower bound and the number of missing entries per row of the incomplete matrix. Further, we develop polynomialtime algorithms for maximum diameter three, which are based on Deza's theorem from extremal set theory. On the negative side we prove NPhardness for diameter at least four. For the parameter number of missing entries per row, we show polynomialtime solvability when there is only one missing entry and NPhardness when there can be at least two missing entries. In general, our algorithms heavily rely on Deza's theorem and the correspondingly identified sunflower structures pave the way towards solutions based on computing graph factors and solving 2SAT instances.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.05068
 Bibcode:
 2020arXiv200205068K
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics;
 F.2.2