Locally Defined Independence Systems on Graphs

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 $α+2k -2$ and $αk$, where $α$ is the approximation ratio of local oracles. The second one can be generalized to the hypergraph setting. We also propose an $(α+ k)$-approximation algorithm for bipartite graphs, in which the local independence systems in the one-side of vertices are $k$-systems with independence oracles.