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arXiv · 2002.05068

The Complexity of Binary Matrix Completion Under Diameter Constraints

Tomohiro Koana·Vincent Froese·Rolf Niedermeier
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Abstract

We thoroughly study a novel but basic combinatorial matrix completion problem: Given a binary incomplete matrix, fill in the missing entries so that every pair of rows in the resulting matrix has a Hamming distance within a specified range. We obtain an almost complete picture of the complexity landscape regarding the distance constraints and the maximum number of missing entries in any row. We develop polynomial-time algorithms for maximum diameter three based on Deza's theorem [Discret. Math. 1973] from extremal set theory. We also prove NP-hardness for diameter at least four. For the number of missing entries per row, we show polynomial-time solvability when there is only one and NP-hardness when there can be at least two. In many of our algorithms, we heavily rely on Deza's theorem to identify sunflower structures. This paves the way towards polynomial-time algorithms which are based on finding graph factors and solving 2-SAT instances.

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Tomohiro Koana, Vincent Froese, Rolf Niedermeier. 2020-02-12. The Complexity of Binary Matrix Completion Under Diameter Constraints. https://arxiv.org/abs/2002.05068

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