Parameterized Low-Rank Binary Matrix Approximation
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We provide a number of algorithmic results for the following family of problems: For a given binary m× n matrix A and 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 k)·(nm)^O(1) and thus is fixed-parameter tractable parameterized by k. We prove that the problem admits a polynomial kernel when parameterized by r and k but it has no polynomial kernel when parameterized by k only unless NP⊆ coNP/poly. We also complement these result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2^O(r·√(k(k+r)))(nm)^O(1), which is subexponential in k for r∈ O(k^1/2 -ϵ) for any ϵ>0. - 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 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·√(kk))(nm)^O(1), which is subexponential in k. - 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·√(k k))(nm)^O(1).
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