The complexity of binary matrix completion under diameter constraints
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 land...
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| Veröffentlicht in: | Journal of computer and system sciences Jg. 132; S. 45 - 67 |
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01.03.2023
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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 (1973) [11] 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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| AbstractList | 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 (1973) [11] 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. |
| Author | Niedermeier, Rolf Koana, Tomohiro Froese, Vincent |
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| Keywords | Stringology Complexity dichotomy Combinatorial algorithms Hamming distance NP-hard problems Consensus problems Graph factors 2-SAT Sunflowers |
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| SubjectTerms | 2-SAT Combinatorial algorithms Complexity dichotomy Consensus problems Graph factors Hamming distance NP-hard problems Stringology Sunflowers |
| Title | The complexity of binary matrix completion under diameter constraints |
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