Complexity of Combinatorial Matrix Completion With Diameter Constraints
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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, upper-bounding 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 polynomial-time solvable and NP-hard cases in terms of the minimum distance lower bound and the number of missing entries per row of the incomplete matrix. Further, we develop polynomial-time algorithms for maximum diameter three, which are based on Deza's theorem from extremal set theory. On the negative side we prove NP-hardness for diameter at least four. For the parameter number of missing entries per row, we show polynomial-time solvability when there is only one missing entry and NP-hardness 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 2-SAT instances.
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