Approximation Schemes for Low-Rank Binary Matrix Approximation Problems

We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are \textsc{Low GF(2)-Rank Approximation}, \textsc{Low Boolean-Rank Approximation}, and various versions of \textsc{Binary Clustering}. For example, for \textsc{Low GF(2)-Rank Approximation} problem, where for an $m\times n$ binary matrix $A$ and integer $r>0$, we seek for a binary matrix $B$ of $GF_2$ rank at most $r$ such that $\ell_0$ norm of matrix $A-B$ is minimum, our algorithm, for any $\epsilon>0$ in time $ f(r,\epsilon)\cdot n\cdot m$, where $f$ is some computable function, outputs a $(1+\epsilon)$-approximate solution with probability at least $(1-\frac{1}{e})$. Our approximation algorithms substantially improve the running times and approximation factors of previous works. We also give (deterministic) PTASes for these problems running in time $n^{f(r)\frac{1}{\epsilon^2}\log \frac{1}{\epsilon}}$, where $f$ is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting in its own.