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# Parameterized Low-Rank Binary Matrix Approximation

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## Cite As

Fedor V. Fomin, Petr A. Golovach, and Fahad Panolan. Parameterized Low-Rank Binary Matrix Approximation. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.53

## Abstract

We provide a number of algorithmic results for the following family of problems: For a given binary m x n matrix A and a nonnegative integer k, decide whether there is a "simple" binary matrix B which differs from A in at most k entries. For an integer r, the "simplicity" of B is characterized as follows. - Binary r-Means: Matrix B has at most r different columns. This problem is known to be NP-complete already for r=2. We show that the problem is solvable in time 2^{O(k log k)}*(nm)^O(1) and thus is fixed-parameter tractable parameterized by k. We also complement this result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2^{O(r^{3/2}* sqrt{k log k})}(nm)^O(1), which is subexponential in k for r in o((k/log k)^{1/3}). - Low GF(2)-Rank Approximation: Matrix B is of GF(2)-rank at most r. This problem is known to be NP-complete already for r=1. It is also known to be W[1]-hard when parameterized by k. Interestingly, when parameterized by r and k, the problem is not only fixed-parameter tractable, but it is solvable in time 2^{O(r^{3/2}* sqrt{k log k})}(nm)^O(1), which is subexponential in k for r in o((k/log k)^{1/3}). - Low Boolean-Rank Approximation: Matrix B is of Boolean rank at most r. The problem is known to be NP-complete for k=0 as well as for r=1. We show that it is solvable in subexponential in k time 2^{O(r2^r * sqrt{k log k})}(nm)^O(1).

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Combinatorial algorithms
• Theory of computation → Fixed parameter tractability
##### Keywords
• Binary matrices
• clustering
• low-rank approximation
• fixed-parameter tractability

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