Parameterized Low-Rank Binary Matrix Approximation

TL;DR

This paper introduces fixed-parameter algorithms and complexity results for binary matrix approximation problems, focusing on low-rank and low-mean structures, with implications for computational efficiency.

Contribution

It provides new fixed-parameter tractable algorithms and kernelization results for binary matrix approximation problems parameterized by rank and error.

Findings

Algorithms for binary r-means problem with fixed-parameter tractability.

Polynomial kernelization for certain parameterizations.

Subexponential algorithms for GF(2)-rank and Boolean-rank approximations.

Abstract

We provide a number of algorithmic results for the following family of problems: For a given binary m\times 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\log k)}\cdot(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\subseteq 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\cdot \sqrt{k\log{(k+r)}})}(nm)^{O(1)}, which is subexponential in k for r\in O(k^{1/2 -\epsilon}) for any \epsilon>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}\cdot \sqrt{k\log{k}})}(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\cdot \sqrt{k\log k})}(nm)^{O(1)}.