  We construct a new class of maximal acyclic matchings on the Salvetti complex
of a locally finite hyperplane arrangement. Using discrete Morse theory, we
then obtain an explicit proof of the minimality of the complement. Our
construction provides interesting insights also in the well-studied case of
finite arrangements, and gives a nice geometric description of the Betti
numbers of the complement. In particular, we solve a conjecture of Drton and
Klivans on the characteristic polynomial of finite reflection arrangements. The
minimal complex is compatible with restrictions, and this allows us to prove
the isomorphism of Brieskorn's Lemma by a simple bijection of the critical
cells. Finally, in the case of line arrangements, we describe the algebraic
Morse complex which computes the homology with coefficients in an abelian local
system.
