2 research outputs found
Parameterized Algorithms for Matrix Completion with Radius Constraints
Considering matrices with missing entries, we study NP-hard matrix completion problems where the resulting completed matrix should have limited (local) radius. In the pure radius version, this means that the goal is to fill in the entries such that there exists a "center string" which has Hamming distance to all matrix rows as small as possible. In stringology, this problem is also known as Closest String with Wildcards. In the local radius version, the requested center string must be one of the rows of the completed matrix.
Hermelin and Rozenberg [CPM 2014, TCS 2016] performed a parameterized complexity analysis for Closest String with Wildcards. We answer one of their open questions, fix a bug concerning a fixed-parameter tractability result in their work, and improve some running time upper bounds. For the local radius case, we reveal a computational complexity dichotomy. In general, our results indicate that, although being NP-hard as well, this variant often allows for faster (fixed-parameter) algorithms
Complexity of Combinatorial Matrix Completion With Diameter Constraints
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