  Given a universe $U$ of $n$ elements and a collection of subsets
$\mathcal{S}$ of $U$, the maximum disjoint set cover problem (DSCP) is to
partition $\mathcal{S}$ into as many set covers as possible, where a set cover
is defined as a collection of subsets whose union is $U$. We consider the
online DSCP, in which the subsets arrive one by one (possibly in an order
chosen by an adversary), and must be irrevocably assigned to some partition on
arrival with the objective of minimizing the competitive ratio. The competitive
ratio of an online DSCP algorithm $A$ is defined as the maximum ratio of the
number of disjoint set covers obtained by the optimal offline algorithm to the
number of disjoint set covers obtained by $A$ across all inputs. We propose an
online algorithm for solving the DSCP with competitive ratio $\ln n$. We then
show a lower bound of $\Omega(\sqrt{\ln n})$ on the competitive ratio for any
online DSCP algorithm. The online disjoint set cover problem has wide ranging
applications in practice, including the online crowd-sourcing problem, the
online coverage lifetime maximization problem in wireless sensor networks, and
in online resource allocation problems.
