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 constraints. The new constrained clustering problem generalizes 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 Low GF(2)-Rank Approximation, Low Boolean-Rank Approximation, and various versions of Binary Clustering. For example, for Low GF(2)-Rank Approximation problem, where for an m× n binary matrix A and integer r> 0, we seek for a binary matrix B of GF(2) rank at most r such that the ℓ0-norm of matrix A−B is minimum, our algorithm, for any ε > 0 in time f(r,ε)⋅ n⋅ m, where f is some computable function, outputs a (1+ε)-approximate solution with probability at least (1−1\e). This is the first linear time approximation scheme for these problems. We also give (deterministic) PTASes for these problems running in time nf(r)1\ε2log 1\ε, 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 on its own.