  We give a new algorithm computing local system cohomology groups for
complexified real line arrangements. Using it, we obtain several conditions for
the first local system cohomology to vanish and to be at most one-dimensional,
which generalize a result by Cohen-Dimca-Orlik. The conditions are described in
terms of discrete geometric structures of real figures. The proof is based on a
recent study on minimal cell structures. We also compute the characteristic
variety of the deleted $B_3$-arrangement.
