Abstract
Low-rank binary matrix approximation is a generic problem where one seeks a good approximation of a binary matrix by another binary matrix with some specific properties. A good approximation means that the difference between the two matrices in some matrix norm is small. The properties of the approximation binary matrix could be: a small number of different columns, a small binary rank or a small Boolean rank. Unfortunately, most variants of these problems are NP-hard. Due to this, we initiate the systematic algorithmic study of low-rank binary matrix approximation from the perspective of parameterized complexity. We show in which cases and under what conditions the problem is fixed-parameter tractable, admits a polynomial kernel and can be solved in parameterized subexponential time.
Highlights
In this paper we consider the following generic problem
Given a binary m × n matrix, that is a matrix with entries from domain {0, 1}, A = ∈ {0, 1}m×n, the task is to find a “simple” binary m × n matrix B which approximates A subject to some specified constrains
To see the equivalence of Binary r-Means and problem (A1), it is sufficient to observe that the pairwise different columns of an approximate matrix B such that A − B 0 ≤ k can be used as vectors c1, . . . , cr, r ≤ r
Summary
In this paper we consider the following generic problem. Given a binary m × n matrix, that is a matrix with entries from domain {0, 1}, A = (aij) ∈ {0, 1}m×n, the task is to find a “simple” binary m × n matrix B which approximates A subject to some specified constrains. To see the equivalence of Binary r-Means and problem (A1), it is sufficient to observe that the pairwise different columns of an approximate matrix B such that A − B 0 ≤ k can be used as vectors c1, . Low Boolean-Rank Approximation Input: A Boolean m × n matrix A, r ∈ N and a nonnegative integer k. Our algorithm for Low Boolean-Rank Approximation is based on solving an auxiliary P-Matrix Approximation problem, where the task is to approximate a matrix A by a matrix B whose block structure is defined by a given pattern matrix P. It appears, that P-Matrix Approximation is an interesting problem on its own. P-Matrix Approximation is NP-complete already for the very simple pattern matrix
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
