Minimal homeomorphisms on the locally compact Cantor set are investigated. We prove that scaled dimension groups modulo infinitesimal subgroups determine topological orbit equivalence classes of locally compact Cantor minimal systems. We also introduce several full groups and show that they are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy. These are locally compact versions of the famous results for Cantor minimal systems obtained by Giordano et al. Moreover, proper homomorphisms and skew product extensions of locally compact Cantor minimal systems are examined and it is shown that every finite group can be embedded into the group of centralizers trivially acting on the dimension group.