  The maximization for the independence systems defined on graphs is a
generalization of combinatorial optimization problems such as the maximum
$b$-matching, the unweighted MAX-SAT, the matchoid, and the maximum timed
matching problems. In this paper, we consider the problem under the local
oracle model to investigate the global approximability of the problem by using
the local approximability. We first analyze two simple algorithms FixedOrder
and Greedy for the maximization under the model, which shows that they have no
constant approximation ratio. Here algorithms FixedOrder and Greedy apply local
oracles with fixed and greedy orders of vertices, respectively. We then propose
two approximation algorithms for the $k$-degenerate graphs, whose approximation
ratios are $\alpha +2k -2$ and $\alpha k$, where $\alpha$ is the approximation
ratio of local oracles. The second one can be generalized to the hypergraph
setting. We also propose an $(\alpha + k)$-approximation algorithm for
bipartite graphs, in which the local independence systems in the one-side of
vertices are $k$-systems with independence oracles.
