  We introduce the community exploration problem that has many real-world
applications such as online advertising. In the problem, an explorer allocates
limited budget to explore communities so as to maximize the number of members
he could meet. We provide a systematic study of the community exploration
problem, from offline optimization to online learning. For the offline setting
where the sizes of communities are known, we prove that the greedy methods for
both of non-adaptive exploration and adaptive exploration are optimal. For the
online setting where the sizes of communities are not known and need to be
learned from the multi-round explorations, we propose an `upper confidence'
like algorithm that achieves the logarithmic regret bounds. By combining the
feedback from different rounds, we can achieve a constant regret bound.
