  Consider the following online version of the submodular maximization problem
under a matroid constraint: We are given a set of elements over which a matroid
is defined. The goal is to incrementally choose a subset that remains
independent in the matroid over time. At each time, a new weighted rank
function of a different matroid (one per time) over the same elements is
presented; the algorithm can add a few elements to the incrementally
constructed set, and reaps a reward equal to the value of the new weighted rank
function on the current set. The goal of the algorithm as it builds this
independent set online is to maximize the sum of these (weighted rank) rewards.
As in regular online analysis, we compare the rewards of our online algorithm
to that of an offline optimum, namely a single independent set of the matroid
that maximizes the sum of the weighted rank rewards that arrive over time. This
problem is a natural extension of two well-studied streams of earlier work: the
first is on online set cover algorithms (in particular for the max coverage
version) while the second is on approximately maximizing submodular functions
under a matroid constraint.
  In this paper, we present the first randomized online algorithms for this
problem with poly-logarithmic competitive ratio. To do this, we employ the LP
formulation of a scaled reward version of the problem. Then we extend a
weighted-majority type update rule along with uncrossing properties of tight
sets in the matroid polytope to find an approximately optimal fractional LP
solution. We use the fractional solution values as probabilities for a online
randomized rounding algorithm. To show that our rounding produces a
sufficiently large reward independent set, we prove and use new covering
properties for randomly rounded fractional solutions in the matroid polytope
that may be of independent interest.
