The Boolean Basis Problem and How to Cover Some Polygons by Rectangles

For $S \subseteq \{ 0,1\} ^n $ the Boolean basis (or set basis) problem is to find a minimum size set $B \subseteq \{ 0,1\} ^n $ such that each $s \in S$ is a Boolean sum of vectors from B. This paper examines tractable special cases of this NP-complete problem. In particular a construction preserving tractability is given. The rectangle cover problem—expressing a rectilinear polygon as the union of a minimum number of rectangles—is a main application.