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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Bibliographic Details
Published in:Journal of computer and system sciences Vol. 132; pp. 45 - 67
Main Authors: Koana, Tomohiro, Froese, Vincent, Niedermeier, Rolf
Format: Journal Article
Language:English
Published: Elsevier Inc 01.03.2023
Subjects:
2-SAT
Combinatorial algorithms
Complexity dichotomy
Consensus problems
Graph factors
Hamming distance
NP-hard problems
Stringology
Sunflowers
Stringology
Complexity dichotomy
Combinatorial algorithms
Hamming distance
NP-hard problems
Consensus problems
Graph factors
2-SAT
Sunflowers
ISSN:0022-0000, 1090-2724
Online Access:Get full text
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Summary: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.
ISSN:0022-0000
1090-2724
DOI:10.1016/j.jcss.2022.10.001