On strong exceptional collections of line bundles of maximal length on Fano toric Deligne-Mumford stacks

We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks ${\mathbb{P}}_{\mathbf{\Sigma }}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of ${\mathbb{P}}_{\mathbf{\Sigma }}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of ${\mathbb{P}}_{\mathbf{\Sigma }}$.